ANALYTICAL TREATMENT OF THE THREE-DIMENSIONAL MODEL OF STIMULATED BRILLOUIN SCATTERING WITH AXIAL SYMMETRIC PUMP WAVE

Transcription

1 Joual of Oolos ad Advad Maals Vol. No. mb ANAYTCA TRATMNT OF TH THR-DMNONA MOD OF TMUATD ROUN CATTRNG TH AXA YMMTRC UM AV V.. Vlad V. ab a A. Moofasu su of Aom hyss NR-D.ass ad Th Romaa Aadmy-CA uhas.o.ox MG-6 Romaa a NO uhas-magul.o. ox MG 5 Romaa Aalyal am of h h-dmsoal modl s sd ad dvlod od o gud h gomy omsao ad h sudy of h asvsal ffs ludg h has ojugao fdly of h oks av. Aalyal soluos foud fo Gaussa ad axal symm um bams hh ovd moa ool aams fo h oss ad a as o b usd omaso h h umal suls. Th ossy of h aalyal soluos obad fo Gaussa o-dld ad dld um avs h hos obad fo mo gal um avs ad h h umal suls as sud. Th aalyal suls f ll h s xmal suls. hav also alulad h flvy ad has ojugao fdly. h aal ass hs moa aams a b as sml fuos of h um gy a b lad a ma hh s ll suod by h umal smulao ad xmal daa. Rvd July 8 ad July Kyods: mulad llou sag Axal symm um av. oduo a b modlld by a s of h dffal quaos dsbg h ao b h lgh avs ad h aousal av yldd a ola mdum [-]. Du o h omlxy of h ola dffal quaos of h h-dmsoal modl sval auhos hav solvd hm dff aoxmaos o umally gvg fomao abou h voluo sa ad m of h h ag avs: um av oks av ad aous av. Rdly al. [] sudd h h-dmsoal oss h h um sy ofl s Gaussa hy hav obad a aalyal soluo h as of quas-sady sa ad odld um ad umal suls h as of dld um. u ad Falk [] ad Mll al. [] hav do umal smulaos of D ad D sady sa ad o-dld um aoxmaos vald a h hshold oly. Numal modls fo D dld sady sa sd by Tag [] ad by Ml ad hl [5]. D dld sady sa as umally sudd by Kummo [6] Moo al. [7] ho dvlod h lgh avs ms of Hm-Gauss ohoomal fuos. oddad al. dvlod a aaly modl fo h valuao of h sag oss-so hh dds o h au aual ad dud oufoms [8]. h sudy of h asvs ffs Vsyauskas al. [9] osud a smlfd h-dmsoal modl h yldal symmy ad foud ou a aalyal soluo fo a al odo ak as a ss of Gauss-agu olyomals. Th soluo ms h olao of som asvsal ad logudal ffs fo x. h dd of oks uls duao o h dvg agl ad h dd of flvy o h asvsal vlo of h um sy.

2 58 V.. Vlad V. ab A. Moofasu V. Raab al. [] sudd h asvsal ffs oal has ojugao lass h mo. Th xaso Gauss-Hm mods s fomd ad h asvsal oblm s lad fo h lay of h soluos. V. ab al [] ad V. Vlad al. [] hav bul a h-dmsoal av modl fo ad hy hav aalyally ad h oss h as of sloly vayg vlo aoxmao ad sady sa gm. Ku ov ad Foad [] ad Mashkov ad Tmk [] sudd h asvsal gmods oagao dff avgudg suus. Ra al. usd [5] a umal modl fo oal fbs h o-ufom os hh u du saal oufoms of h dud aous fld. Akv al[6 ] hmbg [7] Hu al.[8] sudd umally D sady sa h dld um oal fbs h l flds a xadd h ss of oho-omal fuos osodg o h oagao mods of h fbs. Rly. Afshaavahd al. [9] hav sd a as h-dmsoal modl of ad hav usd o sudy h has ojugao ad h mod suu of h oks ad um uls sd h ll. Thy ofm h xmal obsvao of uls-sha dd of has-ojugao fdly sd by Da al. []. hs a fd aalyal soluos of h a dld saal Gaussa TM um bam sady sa hh lad o hgh flvy ad has ojugao fdly. Ou soluos a omad h h suls of Rdly al.[] ad h umal h as D umal smulaos of Afshaavahd ad Muh [9] ad a good agm s foud. Th flvy ad fdly a alulad ad h aalyal suls f ll h xmal ad umal daa. Mo gally s aalyal soluos of h h-dmsoal modl h as of um bams h yldal symmy [ ] hh lad o h vous suls h as of h Gaussa um bams.. h a saal Gaussa um bam h a saal Gaussa TM um bam a b ovly dsbd usg sady sa aoxmao um duao > hoo lfmg quaos h m dvavs vash hh lads yldal oodas o: h s h omlx amlud of h um av h oks av g α k g α k s h omlx amlud of α s h la absoo off of h maal ad ξ kx η ky k ξ η a h omald oodas. Cosdg h um av h Gaussa sa ad m ofl: x [ ] h sy of hs aula bu of usd um s: x [ ] h s h um bam dh ad h aabol aoxmao as usd a h og h h mos sgfa amlfao ous. assum ha h sad bam ll hav a smla saal dsbuo ad look fo a soluo of quaos of h fom: x [ ]

3 Aalyal am of h h-dmsoal modl of smulad llou sag 58 h s h sad bam dh. Aoh aoxmao usually m xms ll b also osdd ou alulaos: h k-dffad lo dvg av aoxmao lag bam adus. odug ad h qs. ad a av o h follog quao sysm: 5 h ad. f h qs. 5 lad o: 6 ad f 5 bom:. 7 Fo a abay adal ooda h quao sysm 5 has fou m gals.. fou ola dffal ould quaos hh dsb boh h logudal ffs by ad h asvsal os by :. 8 Cosdg h al odos: o o 9 h sgula soluo fo of h sysm 8 has h fom: o [ ] o

4 V.. Vlad V. ab A. Moofasu 58 [ ]. h a h dd quaos. Dvlog h ola quao sysm Taylo ss h vy of h al odos o a oba:. Th haas dma of h lad quao sysm s o hus h fou fuos a o la dd h al odos o. hoos h dd fuos of hs sysm ad h dd quaos:. Th haas quao of h sysm : λ λ has h soluos: 9 os π λ 9 os π λ 9 os π λ. hs odos h soluos of h quao sysm ak h fom: 5 h h follog oaos a usd: 9 os π 9 os π ad 9 os π. Th voluo of h um a b dvd fom: 6 as:. 7 od o oba f soluos 5 mos h follog odo o h gao osas:

5 Aalyal am of h h-dmsoal modl of smulad llou sag Usg h al odos o a ddu: s s s s. 9 Usg hs suls a alula a ovall aam hh s sml o b masud h gy flvy: R h a h gs of h um subs ad oks uls subs svly. h as of k dffao h sy of h um fld h Gaussa saal ad moal ofls a b as:. h s h ak sy a s h um bam adus ad s h las uls duao. Th sy of h oks bam may b osdd also as odu of saal ad moal faos h h sgh of h ola oulg s small:... f... f h... f s a fuo lad o h has ojugao fdly f h fuo... f s lad o h las uls omsso f. h ahs h sady sa dfd by us h sauao gm [5] a osd ha h fdly s los o o ad almos osa a df la ad ha h las uls omsso s glgbl h f. Th osodg gs ad h flvy may b as: o d d π ρ ρ R. ould dfy h fs fao o a asvsal omo of h flvy R as. Thus s moa o mak ha h flvy shos a dff dd o h um gy s ad sg a f h s uls omsso o f h s o uls omsso. h sady sa h h uls omsso s alays los o uy h dd of R o h um gy s almos sauad ad ssally dmd by h asvs faus of h oks av. h as of glgbl um uls omsso hh s xd fo log um ulss h flvy a b fuh valuad by:

6 586 V.. Vlad V. ab A. Moofasu R. Af h alulao of h gao osas a fuh oba: h[.85 k l φr ] R R R 5 h[.85 k l φr ] R h: R [ ] ϕ H ϕ R R R h small sgal o flvy h um o-dlo gm o a odu: g << ϕ R ϕ R R R ad h flvy a aks h sml fom: R 6 hh shos h xd la dd o h um sy. mak ha fo um sho uls duao sho ha h hoo lfm h uls omsso ould ou hh mulls h flvy fom 6 by a fao ad lads o h aabol dd: R. h sady-sa sauao a fd aga ϕ R ϕ R R R ad h flvy a b xssd as: R. 7 Th umal alulaos of Rdly al [] sho ha h oks bam s aod du o h hgh ga a h of h um bam ad su h s qualy R < fo 7. Ou aalyal suls also hkd xmally usg a Nd:YAG las osllaoamlf sysm h masug ad oulg sysms ad h abo dsulhd a glass ll as ola maal Fg.. Th osllao osss of a Nd:YAG las Q-shd oad usg a okls ll h KD ysal. Th ouu gy fom h osllao as maxmum.mj maxmum volag of o suly a bam a dffao lmd ad h dud sal dh ad a uls h a m duao of 6s. Th amlfd uls as fousd h ovg lss o a ll oag C as ola mdum. A oal solao a Gla sm ad a qua-av la s odud b h amlf ad h ll. Th las as oad a a o a of H. h xm hav usd a las bam dam of 5mm ad a ls h foal lgh of mm hh lad o a bam as of.6mm h foal la. od o masu h flvy a dg o a mo s lad o h bam o g a f bam fo h um gy ad h baksad oks gy. Th gy of h bams s.

7 Aalyal am of h h-dmsoal modl of smulad llou sag 587 masud usg yol dos RJ-75 ombao h a moo RJ-7 as so ad h uls duao usg a fas hoodod ad a Tkox oslloso TD 5 y. Fg. h dd of h flvy R fuo of h um gy s sho. Th flvy ass h h gy of h d lgh uls sauag a aoxmaly 9%. Th hshold gy s smad a.mj by xaolag h xmal uv of h flvy. Th xmal daa a obad u o h gy osodg o h bakdo gy h abo dsulhd a usual hmal uy. O a o ha a good agm b h alulad flvy ad h xmal daa s obad fo boh h small sgal flvy ad h sauao gms oly. Nd:YAG amlf Nd:YAG osllao Do λ Gla ola λ Do llou ll Fg.. xmal su fo h masum of flvy. R mj Fg.. Th dd of h flvy R fuo of h Gaussa um gy ludg um dlo. h small sgal go h la dd dvd q. 6 s da h ouous l ad h sauao go h aalyal dd fom q. 7 s usd o f h xmal daa makd by os. Th fdly of h has ojugad oks av a b dvd fom h dfo gv by Zldovh al []: H d Σ. 8 d d Σ Σ h ou aalyal soluo fom 5 s ossbl o alula: H H 9 hav dod: h[.85 k l φh ] hlφ h[.85 k lφ ] H H. 9

8 588 V.. Vlad V. ab A. Moofasu H ϕ H ϕ H ϕ H. a valua h fdly h small sgal um o-dlo g << ϕ H ϕ H ϕ H go ad fo as: [ ][ ] H {[ ] [ ] } hh s small ha fo < ad ould ah h [ ] [ ]. h sady sa sauao hav foud: ϕ H ϕ H ϕ H ad h fdly fom 9 aks h fom: H. [ ] [ ] Takg o aou h lao b h bam s ao ad h flvy fom 7 a fuh fd h sml lao: R H R < R hh shos ha fdly gos o fas ha flvy ad ds o as R. Ths dd s sho Fg H R Fg.. Th dd of h fdly H o h flvy R fo Gaussa um avs ludg um dlo. Ou aalyal suls a agm h h bs smulaous xmal suls fo flvy ad has-ojugao fdly obad by Da al.[] h a sglfquy TM oo Q-shd Nd:YF las λ 5m 5s lqud abo ahlod τ s ad gasous N a 9am τ 5s hh a dod h follog abl R x ad H x svly. [ mj ] Tabl R x H x H R

9 Aalyal am of h h-dmsoal modl of smulad llou sag 589 h Tabl h fdly H s alulad h h q. usg h valus R x ad R s alulad h ou q.7 fo h sauao gm. Th hoal daa oma favouably o h xmal os H x. O mak ha o h flvy s masud h fdly ould b smad by q. a las h h um bam duao s muh lag ha h hoo lfm ad h fdly fluuaos a small. Thus ould d h fdly ou xm dvod o h masum of flvy. Usg h flvy daa vsus um gy fom Fg. hav alulad h fdly H h q.. Tabl [ mj ] R x H hs xm h fdly ould ah vy hgh valus du o h hgh flvy. Ths sul s also a good agm h h umal alulaos of Afshaavahd ad Muh [9]. Fo sadad xmal odos h a um uls of s.5m ad gy mj ad fo a maal h h fav dx ad hoo lfm.85s h alulad ovall flvy ahs 78% a sauao ad h fdly 9%. Ou q. ad h abov flvy lad o H.98. Usg h umal alulad daa fom Fg.7 of [9] dod by R sm ad H sm svly hav hkd aga ou aalyal lao b flvy ad fdly fom. Th dos fo h fdly a dod by H ad a sho Tabl. Tabl. h R sm H sm H O ould mak ha h agm of ou do fo fdly q. holds ll h h umal alulaos v h o-saoay fo Gaussa um avs.. Aalyal soluo of h D modl h a saal axal symm um od o mhass h asvs ffs a h av quao hh dsbs h voluo of h lal fld of h lgh bam sd of h ola mdum as: π N α x y h s h fao dx h sd of lgh vauum - h amlud of h oal fld α - h la oal losss ad N - h ola olaao of h mdum. Th al odos fo h lal fld ad s dvav a: x y a df h fuo: x y.

10 59 V.. Vlad V. ab A. Moofasu N π α F. 5 x y Fo gulad dsbuos h gal soluo of q. h al odos a b h fom: τ x y d F x y ξ τ τ dξ 6 τ N h: π F x y α d. 7 x y a odu h logudal haass oodas of h sag oss as: ξ ξ 8 h h mag of h dos alog hh h um ad h oks flds a oagag svly. h asfomaos 8 h soluo 6 boms: τ ξ ξ ξ 9 x y ξ ξ dτ F x y ξ τ dξ τ ξ Th dvavs of h lgh lal fld alog h um ad sad oks haass ξ ξ a svly: x y ξ ξ ξ ξ ξ τ x y ξ ξ Fξ τdτ ξ ad a b dvlod h follog ma: α π N D ξ ˆ ξ ξ π ξ α π ξ ξ τ F ξ τdτ N ξ ξ π h ˆ s h asvs ala oao ad D D a h gao x y osas. qs. a obad fom h av quaos hybol y os by gao of h soluo alog h haass. Thy may b hough as dvg fom Fokk-lak y quaos by a sal avagg. Fo Gaussa osss o a ak D D. od o smlfy h quaos a omal h amlud of h um oks ad aous flds as: πγ ρ 8π 8π a ρ h h maxmum um sy γ h losv off ρ - h dsy of h oagao mdum ad ρ - h dsy vaao du o h ao h h las oal fld. Fuhmo odu h hass of h um oks ad aous flds: ϕ ω k ϕ ω k ad ϕ ω k h ω - h agula fquy of h osodg avs ad k h osodg av vos. Th gal quao sysm ludg asvs ffs aks h fom alog h h haass ϕ ϕ ϕ : D ˆ

11 Aalyal am of h h-dmsoal modl of smulad llou sag 59 ϕ ϕ a α α ϕ ϕ D ϕ ˆ a D ϕ ˆ a A ˆ a g D ϕ a ϕ h α αk αk α k a h omald losss A ω Γ s h ga of h aous fld ω g g γ s h omald llou ga g s h llou ga ρ v Γ π v ω - h mmum ao lgh ad v - h hysoud vloy. Th hd quao of sysm dsbs h voluo of h aousal fld a alog h haas l ϕ. s ddud h sam ay as h quaos fo h oal flds bu fom h Nav-oks quao. ϕ o h Usg h mhod dvlod by Yav [] a oj h haass haas d ϕ : d ω dϕ dϕ ω dϕ dϕ. 5 ω ω shall osd h gomy h yldal axal symmy h h asvs ala oao boms: ˆ h by s dod h omald adal ooda. odu h absoo h flds as: α" ϕ α" ϕ ad 6 od o smlfy h quao g o: ϕ ϕ h α α ω ω α " ϕ A α" g D a ϕ a α " ϕ A α" g Dϕ s a a D a a A ϕ ϕ Dϕ a g α" ϕ a Dϕ a Dϕ a ". Fo ak aousal fld a << h quao sysm 7 a b h fom : ϕ ϕ D [ " ] g α ϕ ϕ ϕ ϕ α" [ ] D ϕ g ϕ ϕ a 7 ϕ. 8 ϕ gag qs. 8 alog h aousal haas do ω k o a oba h quaos of voluo fo h asvsal um ad oks fld omos dod T ad T : a D

12 59 V.. Vlad V. ab A. Moofasu o mo omaly: h g D D ϕ ω x α ϕ ϕ ω k. T g α ϕ T T α ϕ ϕ g 9 T T T T T T Th gao alog h logudal haass ϕ T T T 5 ϕ lads o h doulg of h logudal dvav 8. Th quaos 9 ad 5 hav as bouday odos h logudal soluos foud by us h D modl [5]. Cosdg h asvsal ffs as a oo of h logudal os as as also do by Ml ad hl [5] ould oba h asvsal soluos of h oss. Ths quao sysm a b dud o h dffal quao sysm of h fs od: T T T T T T 5 h ad a h m gals of h sysm 5 hh dsb h sgh of h ola oulg b h o flds. Th slo of hs m gals ould ovd h gal soluos fo h axal-symm um ad oks avs. aulaly akg a ak oulg of h flds by h slo of h fs g umbs fo h m gals: 5 qs. 5 bom: T 5 T T T hh hav as aam h llou ga oly. Takg o aou h oo of h axal losss a fd h av ss: T T x α ω k x x α ω k. ω k ω ω k ω a mak ha h soluo 5 s vald fo ay moal um dsbuo 5 ad h sld gao osas ould dsb h o-saoay small flvy Gaussa as oly. od o hk h soluo 5 a alula aga h flvy: R f 55

13 Aalyal am of h h-dmsoal modl of smulad llou sag 59 h a h gs of h um subs ad oks uls subs svly. Assumg ha h s o las uls omsso h flvy R Fg.. Th flvy vs. omald um gy h small sgal gm fo h sml axal-symm um bam fom q.5 ad R. a small um sy ddud fom h soluos 5 s: f [.7 ] x[.9 ] Ras R R π 56 h.7 <<. 57 Ths small sgal flvy s sho Fg. as a fuo of h um gy ad s agm h h vous suls h Gaussa bams ad h h xmal daa.. Colusos Aalyal h-dmsoal modls sd ad dvlod. Ths modls a moa fo h gomy omsao ad h sudy of h asvsal ffs ludg h has ojugao fdly of h oks av. Aalyal soluos foud hh ovd moa ool aams fo h oss ad a as o b usd omaso h h umal suls. Th ossy of h aalyal soluos obad fo Gaussa o-dld ad dld um avs h hos obad fo mo gal um avs ad h h umal suls as sud. Th aalyal suls f ll h s xmal suls. od o hk h hoal modls hav also alulad h flvy ad has ojugao fdly. h aal ass hs moa aams a b as sml fuos of h um gy. Moov hav ovd ha hy a b lad a ma hh s ll suod by h umal smulao ad xmal daa. Rfs []. Ya. Zl dovh N. F. lsky V. V. hkuov g-vlag l 985 ov. hys. Us [] A. Yav Quaum los d d. Joh ly N.Y []. Kas M. Ma as Hadbook dd by F. T. Ah [] D. T. Ho O. g [5] V. ab A. Moofasu V. Vlad M. J. Dam J. O. o. Am o. vol

14 59 V.. Vlad V. ab A. Moofasu [6] V. T. Tkhohuk C. abau H. A. alads hyss of lasmas [7] R.. Gao H. X. Vu hyss of lasmas [8] abau H. A. alads V. T. Tkhohuk uohyss s [9] V. ouh. gad J Zmmou O. Commu [] V.. Vlad M.J. Dam V. ab A. Moofasu NO ss uhas. [] K. D. Rdly D. C. Jos G. Cook A. M. o J. O. o. Am []. u J. Falk J. O. o. Am []. J. Mll M. D. kldo R.. oyd Al. O [] C.. Tag J. Al. hys [5] R. Ml H. J. hl hys.rv. A [6] A. Kummo O. Commu [7] T. R. Moo R.. oyd J. Nol. O. hys. Ma J. Mod. O [8]. R. odda J. C. Cohus A. G. vy J. D. Coms J. O. o. Am [9] V. Vsyauskas. Gayjauskas. Ghyuas ass ad ulasho osss Russa [] V. Raab A. Hu J. hulhss N. Hodgso J. Kuhs R. Ml Chaos olos ad Faals [] V. ab A. Moofasu V. Vlad o. 5. [] V. Vlad V. ab A. Moofasu H. J. hl Thal Dgs COuo-QC Muh.5. []. A. Ku M.. ov A. A. Foad M. Go D. oh ds. g Vlag [] V. A. Mashkov H. Tmk J. Quaum lo [5]. Ra. o M. J. Cadll O. Commu [6]. Y. Akv. G. Zubav..Mkhalov ov. J. Quaum lo [7] R. H. hmbg J. O. o. Am [8]. H. Hu J. A. Goldso. Ma J. O. o. Am [9]. Afshaavahd J. Muh J. Nol. O. hys. Ma. -7 hys. Rv. A [] C.. Da. A. Numa. A. Hakl O