FUNDAMENTAL AND SECOND HARMONIC AMPLITUDES IN A COLLISIONAL MAGNETOACTIVE PLASMA UDC B. M. Jovanović, B. Živković

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1 FACTA UNIVERSITATIS Sris: Physics, Chmistry and Tchnlgy Vl., N 5, 3, pp FUNDAMENTAL AND SECOND HARMONIC AMPLITUDES IN A COLLISIONAL MAGNETOACTIVE PLASMA UDC B. M. Jvanvić, B. Živkvić Dpartmnt f Physics, Mdical Faculty, Univrsity f Niš, P.O.B. 174, 18 Niš, Srbia and Mntngr Abstract. W prsnt a thrtical invstigatin f frquncy dubling f lctrmagntic wav in hmgnus, cllisinal and magntizd plasma. Th cupld nnlinar quatins fr fundamntal rdinary wav and scnd harmnic xtrardinary wav hav bn slvd. Th amplituds f ths wavs hav bn calculatd fr varius valus f cllisinal frquncy and distanc frm th plasma bundary. 1. INTRODUCTION Th prcss f frquncy dubling f an lctrmagntic wav in plasma has bn invstigatd by numrus authrs [1-7]. Th rasns fr it li in plasma diagnstics, astrphysical rsarch, prblms cnnctd with fusin xprimnts tc. Th influnc f lctrn-in cllisins n th prcss has bn usually nglctd. In this papr, w slv analytically and numrically th prblm f th scnd harmnic gnratin by th rdinary md, prpagating thrugh a hmgnus magntizd, cllisinal plasma. Th dpndnc f th amplituds f th fundamntal wav and its scnd harmnic n cllisinal frquncy and slab thicknss is shwn whn th -phas synchrnism cnditins ar satisfid, i.. whn N =N (), whr N and N () ar rflctin indics fr th fundamntal rdinary and scnd harmnic xtrardinary wavs, rspctivly.. BASIC EQUATIONS W cnsidr plasma in cnstant xtrnal magntic fild B = B z. Th incming lctrmagntic wav prpagats alng th x axis. Th standard nnlinar quatin dscribs th varying f th lctric fild amplituds [5]: Rcivd Fbruary 13, 3

2 46 B. M. JOVANOVIĆ, B. ŽIVKOVIĆ ( ) ( ) ( ) ( ) η η η η η η ω iηωωη [ E ] ε E = c c ε whr η = 1, rfrs t th pump and frquncy-dubld wav, rspctivly. Th (η) dilctric tnsr ε has th frm: whr: and ( η) ε ( ε1 = -iε η) iε 1 ε ε 3 v( η is) v(u) v ε1 = 1, ε =, ε3 = 1 η[( η is) u] η[( η is) u] η( η is) ω p ω v =, u = c v,s = i. ω ω ω Hr ω p and ω c ar th lctrn plasma and lctrn cycltrn frquncis, rspctivly and ν i is th lctrn-in cllisin frquncy. Th pratr (η) is givn in th frm: jnl () ( ), ( () = ik + x = ik + ) x. (3) Th nnlinar lctric currnts n th right-hand sid f quatin ar: () () () m jnl = iωε (1 ε )[ v B + v B ( v ) v ] + () () + n v + n v (4) whr () () m jnl = i ωε (1 ε )[ v B ( v ) v ] + n v (5) ( η ) ε ( η ) ( η ) ( ) ( ) ( ) ( ) 1 ( ) ( ), η ηωε η η η η n = i k E v = i (1 ε η ) E, B = k E. n ηω Taking in accunt rlatins (), (3), (4) and (5) n can btain frm th fllwing st f cupld quatins:

3 Fundamntal and Scnd Harmnic Amplituds in a Cllisinal Magntactiv Plasma 47 ( 1) E ( 1) z = (C 11 ic 1 )E z E y, ( ) E y ( 1) = (C C )E 1 i z, whr C 11, C 1, C 1 and C ar th cupling cnstants, dpnding n u, v and s in th fllwing way: (6) 1 C11 = {mc (N )(1 + s )[(8 s v - u) + s (v 8) ]} () () {s(n )(6s vs 8 u) (N N κ ) () () [8 v u s (v 6)] + s(n κ N κ )(v + s + u)} () ω( v u xp c 1 C1 = {mc (N )(1 + s )[(8 s v u) + s (v 8) ]} () () {(N κ N κ )[8 v u s (v 6)] + + s (N )(16 + s + u v) + () () + s(n N κ )(v + s + u)} () ω( v u xp c 1 C1 = {mc (N )(1 + s )[(8 s v u) + s (v 8) ]} () () [s(n N κ )(16 s u 3v) + () () + (N κ N κ )(8 + 6s u v s v)] v () ω( u xp c (7) C = {mc (N [s(n + (N v () κ () + κ N )(1 + s κ () () )[(8 s )(16 s v u) u 3v) + N κ )(8 + 6s u v s () ω( u xp c + s v)] (v 8) ]} Th cmplx amplituds E z and E y () can b xprssd thrugh th ral amplituds A 1 and A y : 1

4 48 B. M. JOVANOVIĆ, B. ŽIVKOVIĆ z iϕ 1 () y 1 E = A, E = A. y iϕ (8) Nw, th gnralizd phas is intrducd: ω Nx Ψ = ϕ1 ϕ +. (9) c A A y =, (1) + 1 whr C 31 is th functin f u, v and s. On th basis f quatins (6), (8), (9) and (1) th fllwing systm f quatins is btaind: da1 C Ψ Ψ = 11cs C1sin A1A dx + 1 da = + 1( C1csΨ + C sin Ψ) A1 dx dψ A C = C C A csψ dx A + 1 A C C C A sinψ A + 1 If w cmpar this systm with that which crrspnds t cllisinlss plasma [8] w can stablish th diffrncs: 1) first tw quatins in (11) hav additinal trms with cs Ψ, ) th third quatin in (11) includs amplituds A 1 and A. Using th Rung-Kutt [9] mthd th slutins (A 1, A and Ψ) f th systm (11) ar btaind. Fr x = th phas and th amplituds ar givn by: i A1 (11) S π () = sin Φ, A() =, Ψ = (1) c (1 N ) () 1 ε + + κ whr S i is th incidnt nrgy flux and Φ is th angl btwn E () and y axis. 3. NUMERICAL RESULTS Th amplituds A 1 and A ar invstigatd numrically fr paramtrs x and s and th rsults ar plttd in figurs 1-4. Figurs 1 and shw th amplituds f th rdinary fundamntal wav A 1 and th xtrardinary scnd harmnic wav A, rspctivly, against th slab thicknss x fr givn (rsnant) valus f plasma dnsity and xtrnal magntic fild intnsity. Ths figurs dmnstrat that cllisinlss plasma (sris 1) is th mst suitabl fr th scnd

5 Fundamntal and Scnd Harmnic Amplituds in a Cllisinal Magntactiv Plasma 49 harmnic gnratin, bcaus th scnd harmnic amplitud incrass rapidly whil th fundamntal wav amplitud dcrass. Th similar spatial dpndnc f amplituds A 1 and A in lasr plasma (s = 1-4 ) and cllisinlss plasma is vidnt A 1 (V/m) x(m) Sris1 Sris Sris3 Fig. 1. Fundamntal wav amplitud A 1 as a functin f slab thicknss x at varius cllisinal paramtr s. Sris 1, and 3 crrspnd t s = 1 -, 1-4 and 1 - rspctivly. Othr paramtrs: ω =.9GHz, ω p /ω =.8, rsnant valu f ω c /ω, S i = 1 4 W/m, Φ = 45. A (V/m) x(m) Sris1 Sris Sris3 Fig.. Th scnd harmnic wav amplitud A as a functin f slab thicknss x at varius cllisinal paramtr s. Sris 1, and 3 crrspnd t s = 1 -, 1-4, 1 - rspctivly. Othr paramtrs: ω =.9GHz, ω p /ω =.8, rsnant valu f ω c /ω, S i = 1 4 W/m, Φ = 45. Figurs 3 and 4 shw th amplituds f th rdinary fundamntal wav A 1 and th xtrardinary scnd harmnic wav A against th cllisinal paramtr s whn phas synchrnism cnditins ar satisfid. Fr small valus f paramtr s (s < 1 3 ) th amplituds d nt dpnd n th cllisinal frquncy. In plasmas with vry frqunt cllisins (s > 1 3 ) th amplituds attnuat with th incras f paramtr s.

6 5 B. M. JOVANOVIĆ, B. ŽIVKOVIĆ A 1 (V/m) 55 Sris1 Sris Sris lg s Fig. 3. Th fundamntal wav amplitud A 1 as a functin f cllisinal frquncy. Sris 1, and 3 crrspnd t x = 1 m, 6 m and 1 m, rspctivly. Othr paramtrs ar th sam as in figur A (V/m) lg s Sris1 Sris Sris3 Fig. 4. Th scnd harmnic wav amplitud A as a functin f cllisinal frquncy. Sris 1, and 3 crrspnd t x = 1 m, 6 m and 1 m, rspctivly. Othr paramtrs ar th sam as in figur CONCLUSION At th nd f ur papr w can cnclud that cllisins in th magntactiv plasma xrt influnc n th rsnant gnratin f th scnd harmnic amplitud, spcially in plasmas with vry frqunt cllisins (s > 1 3 ). Th influnc f cllisins n this prcss is ngativ, bcaus th valu f th fundamntal wav amplitud dcrass with th incrasing f cllisinal frquncy. REFERENCES 1. Erkhin N.S., Misv S.S. and Mukhin V.V., Nucl. Fusin, 14, 333, (1974).. Stnfl L. and Dysth K.B., J. Gphys. Rs., 83, 7, (1978). 3. Aur G., Saur K. and Baumgartl K., Phys. Rv. Ltt., 4, 1744, (1979). 4. Dragila R., Phys. Rv. A, 8, 361, (1983).

7 Fundamntal and Scnd Harmnic Amplituds in a Cllisinal Magntactiv Plasma Čadž V. M. and Jvanvić D., J. Plasma Physics, 35, 15, (1986). 6. Čadž V. M. and Jvanvić, ASP Cnfrnc Sris, 93, 97, (1996). 7. Jvanvić, B.M., Phisica Scripta, 64 (), 161, (1). 8. Jvanvić B.M., Cntributd Paprs f 18 th SPIG, 564, (1996). 9. W.H. Prss, B.P. Flanrry S.A., Tuklsky and Vttrling, Numrical Rcips, 1, Cambridg Univrsity Prss, (1986). AMPLITUDE OSNOVNOG I DRUGOG HARMONIKA U KOLIZIONOJ MAGNETOAKTIVNOJ PLAZMI B.M. Jvanvić, B. Živkvić Prikazuj s trijsk istraživanj dupliranja frkvncij lktrmagntng talasa u hmgnj, klizinj i magntizvanj plazmi. Ršn su sprgnut nlinarn jdnačin za snvni rdvni i nrdvni talas drugg harmnika. Amplitud vih talasa su računat za različit vrdnsti klizin frkvncij i rastjanja d granic plazm.