Armnian Journal of Physics, 15, vol. 8, issu, pp. 64-7 TREATMENT OF THE PLASMA NONLINEAR ABSORPTION LAW AT LINEARLY POLARIZED LASER RADIATION OF RELATIVISTIC INTENSITIES A. G. Ghazaryan Cntr of Strong Filds Physics, Yrvan Stat Univrsity, 1 A. Manoogian, Yrvan 5, Armnia -mail: amarkos@ysu.am Rcivd 11 March 15 Abstract On th bas of prvious analytical invstigations of invrs lasr-inducd multiphoton brmsstrahlung of lctrons on th ions/nucli in th low-frquncy approximation, and nonlinar absorption cofficint of linarly polarizd lasr radiation of rlativistic intnsitis in plasma is tratd via numrical simulations. Som dpndncs of absorption cofficint on th radiation intnsity for modratly strong, as wll as asymptotically larg valus of short lasr pulss and high tmpraturs of plasma ar rvald in cas of radiation linar polarization. Th lattr xhibits diffrnt absorption law and nonlinar bhavior of plasma absorption cofficint comparing with th cas of circular polarization. Kywords: rlativistic suprintns lasr radiation, plasma, nonlinar absorption, brmsstrahlung. At th propagation of an lctromagntic (. m.) wav in th plasma, on of th dominant mchanisms of radiation absorption is th invrs stimulatd brmsstrahlung (SB). Th inducd absorption procss of strong. m. radiation in plasma du to th multiphoton SB of lctrons on th Coulomb scattring cntrs, in th limit of th low-frquncy approximation, has bn invstigatd with th apparanc of lasr sourcs (for arlir considration of this procss s th rviw [1]). At th ultrahigh intnsitis of supr short lasr pulss considrably xcding nowadays th rlativistic thrshold valu of radiation filds ( 1 18 Wcm in optical domain), th nonlinar absorption procss du to an lctron bam SB on th Coulomb scattring cntrs has bn invstigatd in th papr [], and a nw dpndnc on th wav intnsity ( 1/, F / mc is th dimnsionlss rlativistic invariant paramtr of intnsity,, m th lctron charg and mass, F, th lctric fild amplitud and frquncy of a lasr radiation, c
Tratmnt of th Plasma Nonlinar Absorption Armnian Journal of Physics, 15, vol. 8, issu th light spd in vacuum) of th absorption cofficint of a homognous mononrgtic lctron bam on th wav intnsity at asymptotically high valus ( 1) has bn rvald. As th intraction of suprintns lasr pulss with any mattr rsults to lasr-plasma intraction and sinc th lasr-producd plasmas ar bright sourcs of X-ray mission, thn du to th high lasr absorption at th formation of lasr plasmas, apart from plasma hating and rlatd important procsss or its practical applications, th convrsion fficincy of lasr fild nrgy into th X-ray mission spctrum is significantly high. Bsids th thrmal sourcs of incohrnt X-ray, during th last dcads lasr-producd plasmas ar intnsivly discussd as th activ mdia for gnration of cohrnt X-ray, spcially via high harmonic gnration in th soft X-ray rgion (s,.g. [3]). So, in th currnt stag of invstigations of suprstrong lasr fild-plasma intraction procsss th problm of nonlinar absorption of lctromagntic wav nrgy by plasma bcoms important. As in th strong. m. wav fild SB procss has significantly multiphoton natur, th task can b considrd in th scop of classic thory [1]. First th study of th absorption of strong radiation in fully ionizd plasma in th SB procss was prformd on th basis of th kintic thory in [4]. In this papr w considr th plasma nonlinar absorption dpndnc on th intnsity of xtrnal radiation using xact analytical xprssions [] for absorption cofficint in low frquncy (LF) approximation. Th conditions of th assumd LF approximation and th cas of rlativistic plasma in th circular polarization wav fild hav bn considrd by us arlir [5]. Howvr, it is wll known that th kintics of an lctron in th fild of a strong. m. wav ssntially dpnds on th polarization of th wav. Thus, sinc th wav intnsity for a circular polarization const, thn th longitudinal vlocity of th lctron in th wav: V II const too, manwhil in th wav of a linar polarization V II oscillats with th wav harmonics n, corrsponding to inharmonic oscillatory motion of th lctron. Th lattr lads to mor complicatd bhavior of th dynamics of lctron inducd intraction with additional third body at th linar polarization of a stimulating strong wav. For xampl, in contrast to circular polarization [6], in cas of a linarly polarizd. m. wav to obtain ultimat rsults for rlativistic ATI rats, taking into account th photolctron SB, is impossibl analytically [7]. Th analogous situation taks plac for rlativistic multiphoton SB in th strong linarly polarizd 65
Ghazaryan Armnian Journal of Physics, 15, vol. 8, issu radiation fild, th considration of which is th mattr of th prsnt papr. On th othr hand, many important lasr-assistd procsss and nonlinar phnomna just occur at th linar polarization of th stimulating fild (whn consrvation laws of th procss rquir crtain symmtry of th photon fild). Thus, th high harmonic gnration (HHG) procss on th atoms taks plac only in th linarly polarizd lasr filds [3]. To rval dpndnc on th wav polarization, in this papr w study th nonlinar absorption cofficint in undrdns plasma du to th mchanism of th nonlinar SB of lctrons on th Coulomb scattring cntrs for th linarly polarizd lasr radiation of rlativistic intnsitis. Du to th complxity of th task th invstigations ar prformd by th numrical tratmnt of th issu. For intrmdiat, as wll as at high tmpraturs of lctrons and asymptotically larg valus of lasr filds with rlativistic lasr intnsitis, as it has bn shown in [], th SB procss is wll nough dscribd by th classical thory, in th LF approximation. Hnc, th absorption cofficint for a radiation fild of arbitrary intnsity, in gnral cas of th homognous nsmbl of lctrons with th arbitrary distribution function f (p) ovr rlativistic momntum p, at th invrs-brmsstrahlung on th scattring cntrs with concntration n i can b rprsntd in th form (cm -1 ): n i I d f ( p ) d p W ( p, ) d, (1) dt whr n i W is th classical nrgy absorbd by a singl lctron pr unit tim du to SB procss on th Coulomb scattring cntrs; is th scattring phas in th. m. wav, and I A / 8c is th wav intnsity of linar polarization (th intgration is prformd ovr th initial phas ). For th gnrality, w assum Maxwllian plasma with th rlativistic distribution function of lctrons by momnta: f n ( p) 4m cktk m c / k B T p E xp kt, () 66
Tratmnt of th Plasma Nonlinar Absorption Armnian Journal of Physics, 15, vol. 8, issu whr k B is Boltzmann s constant, T and n th tmpratur and concntration of lctrons in plasma rspctivly, E (p) is th rlativistic nrgy momntum disprsion law of lctrons and x K is McDonald s function; f (p) is normalizd as f ( p) dp. (3) According to th work [], th chang of nrgy of on lctron du to th scattring on Coulomb cntrs (in LF approximation) in th strong. m. wav fild of linar polarization with th vctor potntial: n A( ) A ˆ cos( ), (4) (ê is th unit vctor; ˆ k, k and ar th wav vctor and phas, rspctivly) at a crtain phas is givn by th rlation (taking into account th dfinitions p p ( ), E E ( ) ): dw dt Z m c E np' 4 3 3 ( np' ) p m c a p, 1 E cos Z Z cos p m c np 1 E 3 (5) m c Z Z cos cos Z Z cos ' ln 1 E np' m c 3 cnp' m Z a p E 3, whr E A, p p nc cos Z Z cos (6) c E c cos Z Z cos ar th nrgy and momntum of an lctron in th wav, p' p ne / c, n k/ k. Th paramtrs Z and hav th form: 3 A / 4c np', A pˆ / c ' Z np. (7) Th rlation for th absorption cofficint in th cas of linarly polarizd wav is vry complicatd and vn for not so larg on cannot intgrat it analytically. Thrfor, for th analysis w hav prformd numrical invstigations, making also analytic intrpolation. For th numrical simulations in (1) w hav takn Z 1, = 1 V. Numrical calculation of th a invrs-brmsstrahlung absorption cofficint (1) has bn mad for intrmdiat, as wll as at larg valus of lasr filds and high tmpraturs of lctrons. To show th dpndnc of th 67
Ghazaryan Armnian Journal of Physics, 15, vol. 8, issu invrs-brmsstrahlung absorption rat on th lasr radiation intnsity for modrat wav intnsitis.1 1, in figur 1, th scald rat (, T n) / vrsus plasma tmpratur for various wav intnsitis is shown ( T n k B T / mc 3 a ). Hr, 4Z r n n, (8) whr is th lasr radiation wavlngth, r is th lctron classical radius. For th comparison, in Fig. 1 th numrical rsults for th nonrlativistic absorption cofficint (1) ar also shown. As xpctd for small valus of both rsults coincid, in th mantim, th absorption rat for a modratly larg at th invrs-brmsstrahlung, givn by 3/ th nonrlativistic formula (1), is supprssd du to dpndnc 1/ I [1]. i Fig. 1. Total scald rat of invrs-brmsstrahlung absorption (in arbitrary units) of linarly polarizd lasr radiation in plasma, as a function of th plasma tmpratur (in units of an lctron rst nrgy mc ) is shown for various wav intnsitis: (a). 1, (b). 5 4 1 and (c) 1 for th rang T k T / mc 1 1. Numrical rsults for th n nonrlativistic absorption cofficint [1] ar shown by th dashd lins. B 68
Tratmnt of th Plasma Nonlinar Absorption Armnian Journal of Physics, 15, vol. 8, issu In Fig., th dpndnc of th invrs-brmsstrahlung absorption rat on th lasr radiation intnsity for th linarly polarizd wav is shown for various plasma tmpraturs. As sn from this figur, th SB rat is considrably supprssd with th incras of th wav intnsity and plasma tmpratur. Th rsults of numrical invstigations of quation (1) for th larg valus of lasr filds and high tmpraturs of lctrons ar illustratd in Figs. 3-5. In Fig. 3, th scald rat (, T n) / vrsus th rlativistic invariant paramtr of th wav intnsity for various plasma tmpraturs is shown. As sn from Fig. 3, th SB rat strictly dpnds on th wav polarization; it is supprssd with an incras of th wav intnsity, and for larg valus of it xhibits a tnuous dpndnc on th plasma tmpratur. This bhavior is also sn from Fig. 4, whr th total scald rat of th invrs-brmsstrahlung absorption as a function of th plasma tmpratur T n (in th units of an lctron rst nrgy mc ) is shown for various wav intnsitis. Hr, for th larg valus of, w hav a wak dpndnc on th tmpratur, which is a rsult of th lasr-modifid rlativistic scattring of lctrons, irrspctiv of th initial stat of lctrons. As it was shown in [5], in th cas of circularly polarizd wav th absorption cofficint α dcrass as 1/ at th incras of intnsity, in accordanc with th analytical rsults []. Fig.. Total scald rat of invrs-brmsstrahlung absorption (in arbitrary units) of linarly polarizd lasr radiation in plasma vrsus th dimnsionlss rlativistic invariant paramtr of wav intnsity in th rang.1 1 for various plasma tmpraturs. 69
Ghazaryan Armnian Journal of Physics, 15, vol. 8, issu Fig. 3. Total scald rat of invrs-brmsstrahlung absorption (in arbitrary units) of linarly polarizd lasr radiation in Maxwllian plasma vrsus th dimnsionlss rlativistic invariant paramtr of th wav intnsity for various plasma tmpraturs. Fig. 4. Total scald rat of invrs-brmsstrahlung absorption (in arbitrary units) of linarly polarizd lasr radiation in plasma, as a function of th plasma tmpratur (in units of an lctron rst nrgy mc ) is shown for various wav intnsitis. 7
Tratmnt of th Plasma Nonlinar Absorption Armnian Journal of Physics, 15, vol. 8, issu α/α L.1.4.7 T n.9 ξ Fig. 5. Plot of th total rat of invrs-brmsstrahlung absorption scald to th asymptotic rat L (in arbitrary units), as a function of th plasma tmpratur (in units of an lctron rst nrgy mc ) and th dimnsionlss rlativistic invariant paramtr of th linarly polarizd lasr bam. In th cas of linarly-polarizd wav from Fig. 5 with th intrpolation, w hav sn that 4 dcrass as 1/ 5/ at th incras of th wav intnsity and xhibits a tnuous dpndnc on th plasma tmpratur. For th larg w can intrpolat by th following formula. As sn from Fig. 5, in th cas of linarly-polarizd wav and for th modrat tmpraturs, with th wll nough accuracy on can apply th asymptotic rat (9):. (9) L 5 / 4 Concluding, w hav prsntd numrical tratmnt of rlativistic thory of th invrsbrmsstrahlung absorption of a suprintns lasr radiation in th LF approximation. Th cofficint of nonlinar invrs-brmsstrahlung-absorption in plasma has bn calculatd considring, in gnral, th rlativistic Maxwllian distribution. Th simpl analytical formula (9) has bn obtaind for th absorption cofficint at asymptotically larg valus of lasr filds for linarly polarizd. m. wav. Th obtaind rsults dmonstrat that th SB rat is supprssd with th incras of th wav intnsity and tmpratur of plasma. If for larg valus of, th absorption cofficint dcrass as 1/ for circularly polarizd wav [5], for a linarly 71
Ghazaryan Armnian Journal of Physics, 15, vol. 8, issu 4 polarizd. m. wav dcrass as 1/ 5/, in contrast to th nonrlativistic cas whr on has 3 th dpndnc 1/ [1]. Th SB rat is supprssd with th incras of th plasma tmpratur, but for th rlativistic lasr intnsitis it xhibits a tnuous dpndnc on plasma tmpratur. Acknowldgmnts I would lik to thank Prof. H. K. Avtissian and G. F. Mkrtchian for valuabl discussions during th work on th prsnt papr. This work was supportd by Stat Committ of Scinc of RA. Rfrncs 1. F. V. Bunkin, A. E. Kazakov, M. V. Fdorov, Sov. Phys. Usp. 15, 416 (1973).. H. K. Avtissian, A. K. Avtissian, H. A. Jivanian, S. V. Movsissian, J. Phys. B 5, 317 (199); ibid B 5, 31 (199). 3. T. Brabc and F. Krausz, Rv. Mod. Phys. 7, 545 (). 4. V. P. Silin, Sov. Phys. JETP, 151 (1965). 5. A. G. Ghazaryan, Armnian Journal of Physics 7 (1), 1 (14). 6. H. K. Avtissian, A. G. Markossian, G. F. Mkrtchian, Phys. Rv. A 64, 5344 (1). 7. H. K. Avtissian, A. G. Markossian, Phys. Rv. A 76, 5347 (7). 7