West Virginia University

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1 Ws Viginia Unisiy Plasma Physics Gup Innal Rp PLP-47 Fis d pubd lciy disibuin hy and masumn Jhn Klin Chisian Fanck and Rb Spangl Fbuay 5 Vsin.

2 Tabl Cnns Diniin Tms iii Inducin Elcsaic was. Ingaing lciis 8.. Ingaing 8.. Ingaing..3 Ingaing y..4 Singl Vlciy Cmpnn Elcsaic 5. Cnsain ω k and k using h Dispsin Rlainship 6 3 Elcagnic 8 3. Ingaing lciy cmpnns Ingain Singl Vlciy Cmpnn Elcmagnic 6 4 Las Inducd Flscnc masumn 7 4. asumn wih a Lck-in amplii 7 4. asumn Using Aagd Tim Sis Pssibiliis Using Csspw Spcum 34 APPENDIX A 35 BIBLIOGRAPHY 39 ii

3 Diniin Tms = cycln quncy bh ins and lcns = mass bh ins and lcns ω = wa quncy k = ppndicula wa numb k = paalll wa numb k ds = Dby wa numb spcis s Φ = Elcsaic pnial wa ampliud = ppndicula hmal lciy = paalll hmal lciy = ppndicula wih spc ambin backgund magnic ild = paalll wih spc ambin backgund magnic ild iii

4 Inducin In 987 F Ski and F. Andgg publishd h is pap hiing h pssibiliy masuing wa numbs m a ppagaing wa using Las Inducd Flscnc LIF. 3 Du h nn-lcal nau h plasma dilcic uncin a lcal masumn pids inmain abu h spns h plasma du lcsaic lcmagnic was. Using h lciy and spaial sluin LIF masumns h pubd in lciy disibuin du a wa can b masud. Fm lina Vlas hy h mhd chaacisics 456 can b usd calcula h pubd lciy disibuin uncin ins and lcns. Using h hy i h pimnal daa als h wa numb inmain a wa. Th adanag his mhd as wih all LIF masumns is claly h ac ha h masumn is nn-inusi hus masuing h walnghs was wihu acing h plasma. This dcumn dlps h hical as wll as h pimnal chniqu masuing wa numbs using LIF. Th hy lcsaic was will b ulind in scin II a gnaliain lcmagnic was will b dlpd in scin III scin IV will cnain pimnal dails masuing h pubain h disibuin uncin and scin V will cmmn n h applicains using h masumn. Elcsaic was Saay 7. al. publishd h is pap wih pimnal suls masuing h is d pubain h disibuin uncin and calculaing h wa numbs lcsaic was. This was dn using an annna launch a wa in h plasma and using LIF cllc h inmain abu h wa. Elcsaic was w usd bcaus hy a h asis was launch and ha h asis hy as will b sn la in h discussin lcmagnic was. Saing m h cllisinlss Vlas quain a mhd ingaing h unpubd bis knwn as h mhd chaacisics 456 is usd calcula h is d pubain h lciy disibuin uncin. Sinc h is d pubain h lciy disibuin uncin is usd calcula h h plasma dilcic ns h diain llws clsly h dlpmn h h plasma dilcic ns calculains dn by Swansn 8. Th lina Vlas Equain wih n cllisins is d = d q [ E B ] = This qain is h cnsain paicals in im wh al im diai h si dimnsinal phas spac is

5 wh is h lciy and a is acclain. Th Vlas quain can b linaid in h llwing way 3 and quain can hn b win as 4 H h is m in baks is d /d h h d las quain which is sinc h plasma is asummd b in an quilibium sa. Th scnd m in baks is d /d h is d las quain. N ha sinc is an indpndn aiabl sinc h is n quilibium lciy as in luid hy lciy is n linaid. I w nglc h scnd d m h las m quain 4 bcms 5 Sling quain 5 bcms 6 7 Equain 7 is h gnal m wih h iniial cndiin. I nly was ha gw in im a cnsidd hn will g as. Thus h gnal is d pubain h lciy disibuin uncin is 8 Sinc his scin is nly cncnd abu lcsaic was k B = B can b s. 9 ] [ B E d q = a d d = = ] [ ] [ ] [ ] [ = = = B E q B E q B E q B E q d d d d d d ] [ = B E q d d ] [ B E q d d = E d q = ] [ B E d q =

6 T sl his quain h is sp is wi h lcic ild as an lcsaic aling wa. E i k i k i k = E ω = Φ ω = ikφ ω wh Φ is h ampliud h lcsaic pnial h wa. N a disibuin uncin quain 9 nds b chsn. In a gnal cas any disibuin uncin ha is indpndn im i.. a sluin h sady sa h d Vlas quain can b usd cmpu. Hw in gnal h ms cmmnly usd disibuin uncin is h Bi-awllian wih a di alng h magnic ild as discibd by quain. / / y y = π π / Thus h quain h is d disibuin uncins bcms q = d ikφ i k ω π π / N ingain h unpubd paicl bis is dn. T cmpl his ingal h gnal unpubd quains min a ndd and can b did m h min a chagd paicl in a unim magnic ild wih quain 3. d = ˆ d wh is h cycln quncy and pim dns aiabls ha a uncins h aiabl im. Wih bing h im h iniial cndiins τ is h im bwn h masmn and h iniial cndiins τ = -. Sling hs dinial quains gis h unpubd lciy quains which can b ingad g h unpubd quains psiin. y / / 3 Vlciy quains Psiin quains = cs τ εysin τ ε = sin τ y cs τ ε y = εsin τ y cs τ y = y cs τ y sin τ = = τ 3

7 H ε is q/ q wh q is h chag. This pids h cc dicin ain lcns and ins. Wih hs quains quain can b win in ms τ and h ingain aiabl changd τ. W sa by wiing h pnnial m in ms τ. i i k ω = i k ω ksin τ εky cs τ iy ky sin τ εk cs τ i ω k τ 4 Ging cylindical cdinas using k k y = k cs φ = k sin φ 5 and k = k h pnnial in h lcic ild can b win as εk p[ ik ω ] = p[ ik ω i ycsφ sin φ k k y i sin φ ε τ i sin φ ετ π i ω k ] τ 6 Using h llwing Bssl idniy ib sin θ = n = h pnnial m bcms k J J ik ω ik ω y k = m n mn = J b inθ n εk cs sin i n m k i y inm φ φ ωε τ φ imπ 7 8 Nw ha h pnnial m is win in ms τ h gadin h disibuin uncin can als b win. k k cs yk sin φ φ k = 9 aking h apppia subsiuins in quain 9 h pimd aiabls yilds 4

8 k = which is quialn ε cs τ ε sin τ sin τ k y cs τ k y cs φ sin φ k k k = ε k cs τ cs φ ε sin τ sin φ y sin τ cs φ cs τ sin φ k Using h ignmic idniis h angl addiin and subacin quain can b win as k k cs φ ετ y cs φ ετ π k = Equain and quain 8 a subisiud in quain and h ingain aiabl changd τ using τ = -. Th sul is iqφ = ik ω k y k dτ Jm Jn mn = εk inm imπ i φ i ωε nm k cs sin τ y φ φ φ ε τ π y φ ε τ k cs cs k Nw h ingal is a uncin a singl aiabl τ. Th n sp is wi h csin ms in pnnial m. iqφ k = d J J εk cs sin ik i ω y φ φ y k τ m n mn = i ω ε n m k τ i n m φ imπ k k k i π i π i φ ε τ i φ ε τ φ ε τ φ ε τ y 3 4 Raanging quain 4 by muliplying ach m hugh in h ingal gis 5

9 εk i iqφ cs sin y φ ik φ ω k k = dτ mn = k k J J i ω ε n m k τ in m φ i ω ε n m k τ in m φ imπ y m n y i ω ε n m k τ inm φ im π i ω ε n m k τ inm φ im π y m n k k k J J i ω ε n m k τ i n m φ imπ y m n k J J k 5 N h indics h summains a adjusd duc quain 5. F simpliciy ls wk wih n m as an ampl. k k k i nm k inm im π y ω ε τ φ y J m Jn nm = 6 Nw l p=m- m=p. k k k y i ω ε n p k τ in p φ i p π y J p Jn n p = Sinc h summain is m - and h ms g small as hy g ininiy h sul quain 7 is h sam as k y i ω ε n p k τ i n p φ i p π k y k J p Jn np = Nw n hugh i is cnusing p is s qual m p=m gi k y k i ω ε n m k τ i n m φ i m π y k Jm Jn nm = Applying his chniqu h ncssay ms quain 5 can b win as εk iqφ i cs sin y φ ik φ ω = dτ mn = k k k k y J n Jn Jm k y k y k y k J m Jm Jn k k y k J m Jn i ω ε n m k τ n m φ imπ

10 Nw h Bssl lainship k k Jn = Jn Jn n can b usd wi quain 3 as εk i iqφ cs sin y φ i k φ ω = n m k k k y J m J m n= n i n m φ imπ i ω ε n m k τ dτ 3 3 T d h ingal τ i is assumd ha h quncy ω has a small bu ini imaginay pa ω=ω al iω imaginay. This maks i pssibl d h ingal in quain 3 sinc i gs in h limi. Physically his mans h was ha a ini gwh a sinc hy ha bn assumd gw in im. Thus h ingal is dτ i ωε nm k τ = i ω ε n m k Puing his back in quain 3 gis Φ k J J i k ω y = m n mn imπ / i m n φ iε k y / ky / ε m n k ε n m k ω k Equain 34 is h pubd lciy disibuin uncin and can b und in Saay. al. 7 wih h addiin h p-ik -ω m. Sinc h pubd lciy disibuin uncin is cad by a wa h phas h pubain dpnds n h phas h wa. Thus i yu masu a w din psiins ims in a wa h phas will b din and his phas dinc ppagas as p-ik -ω jus as h wa. This m culd b md and h maining pa hugh as a Fui ampliud such ha = p-ik -ω bu will b l in his dlpmn. 7

11 . Ingaing lciis Th pups his calculain is ind an quain ha can b usd i pimnal LIF daa. Sinc LIF masus nly n cmpnn lciy h quain mus b a uncin a singl cmpnn lciy. T achi his ingain w cmpnns lciy can b dn g a uncin in ms h singl maining cmpnn. Th lciy cmpnns in quain 34 can b spaad as llws Φ i k ω imπ / i m n φ = mn / k y i k / y ε y J m π / iε ky / k J n π 35 π / ε m n k ε n m k ω wh ach m in baks is a uncin nly n cmpnn lciy. Ingaing any w h ms in baks wih spc h lciy aiabl in h bak will la bhind as a uncin h maining aiabl. Th n hing d is inga ach h ms in baks. Thn w can subsiu any w h suls h ms in baks g a uncin h maining aiabl... Ingaing Th ingal is as llws: ε m n k / / d ε ω π π n m k T d his ingal h llwing subsiuin is usd 36 s = d = ds 8 37

12 Using hs lainships h ingal can b win as ε m n ks / s ds π ε n m k ω k s k L ω ε n m k δ n m= k and quain 38 can b win as ε m n s ds π h k s δ s m n Using h ac ha δ =k -ω/ k quain 4 can b win as π ds δ δ n m h s s δ s m n Nw quain 4 is in a m ha can b ingad wih h hlp h plasma dispsin lainships Z δ = Z δ = π π s ds s δ s s ds s δ Z δ = [ δz δ] Using hs lainships quain 4 can b win as δ δ n m Z δ n m Z δ n m δ δ n m Z δ n m δ n m Z δ n m

13 This can b ducd uh using h ac ha / = T /T : d π / ε m n k = ε n m k ω T T δ T T δ n m Z δ n m 47.. Ingaing Th ingal is / d J π k ik y / ε n 48 Fis h dinay Bssl uncin can b win as J n k = π π dα k i sinα inα 49 mak h ingal asi. Wih quain 49 quain 48 can b win as / π ik y / ε π d dα π π k i sinα inα / k π i sinα εsin φ inα dα d π 5 5 Th n sp in ingaing is cmpl h squa h pnnial m. π π / k sin sin k k i ε φ α εsinφ sin α εsinφ sin α d / k k i εsinφ sin α sin sin ε φ α d 5 53

14 U subsiuin is usd cmpl h ingal wih u = d = du k i ε sinφ sin α 54 Nw quain 53 can b win as π sinc h las m in h pnnial has n dpndnc. Wih h aluain h ingal du u / = k sinα sin φ du π u quain 53 bcms π / k k i εsinφ sin α εsinφ sin α k εsinφ sin α d = 57 Subsiuing back in quain 5 gis π π k εsinφ sin α inα dα 58 Th n sp is pand and algbaically ansm h pnnial m wih h llwing idniy λcs τ = I n λ n= inτ Th n s sps ansms quain 58 in a m us wih quain 59. π π k sin φ εsinφsinα sin α inα dα k sin φ π k k sin α α in π dα ε sinφ sinα

15 Using sm ignmic idniis yilds k k sin k cs π φ π α εsinφcs α inα dα π 6 Using h idniy quain 59 quain 6 bcms k k sin π φ sin π inα k i lα εk φ ipα ip π 4 dα I l l= ingal α is cmpld using hgnaliy giing l = n-p/ wh l mus b an ing. Using his quain 64 bcms I p p= k k sin φ ipπ π εk 4 φ k inα ilα ipα p l α p= l= sin I I d π π inα i lα ipα π l p n = dα = l p n Th k sin φ k 4 p= I p εk sin φ I n p k ip π 66 I w din a= kp a= k / and c= k sinφ his can b win as c a 4 8 ac a I ε p In p π p= 8 Thus h ingal is / c a ik y/ 4 8 ε k εac a n = p np π p= 8 ip d J I I ipπ Ingaing y Th ingal y is naly h sam as ha bu b plici h ingain is als dn h. Th ingal y is / dy J π k y ik ε y / y m

16 69 As b dinay Bssl uncin is win as J m k y = π π dα k i sinα imα mak h ingal asi. Wih his lainship quain 69 can b win as / π y ik ε y / y π d dα π π Th n sp in h ingal y is cmpl h squa h pnnial m. π π U subsiuin is usd cmpl h ingal wih y k u = i ε csφ sin α d y ky i sinα imα / ky y cs sin π i ε φ α imα = π dα d y / ky y k cs sin cs sin k i ε φ α ε φ α ε csφ sin α dy / y k k i ε csφ sin α ε csφ sin α dy du Nw quain 74 can b win as π sinc h las m in h pnnial has n y dpndnc. Wih h aluain h ingal du u / = k εcsφ sin α du π u quain 74 bcms π / y k k i εcsφ sin α εcsφ sin α k εcsφ sin α d y = 3

17 Subsiuing back in quain 7 w g π π k ε csφ sin α imα dα Th n sp is pand and algbaically ansm h pnnial m using h llwing idniy λcs τ = I n λ n= inτ 8 Th n s sps ansms quain 79 in a m us wih quain 8. k cs cs sin sin π φ ε φ α α imα dα π k cs k sin k φ cs sin π α ε φ α imα dα π Using sm ignmic idniis yilds k k k cs cs π φ cs cs π α ε φ α imα dα π Nw using quain 8 quain 83 bcms k k cs φ π 4 im k il k α l p l= p= α α ε csφ d I I π k k cs φ π εk cs 4 φ k Ip I l dα π p= l= Th ingal α is dn using hgnaliy π d = l p m imα iαl ipα π l p m α = ipα ipπ π ip imα ilα ipα gis l = p-m/ wh l mus b an ing. Thus quain 85 bcms k k cs k cs 4 k ip φ ε φ I p I p m π p= 4

18 I w din a= kp a= k / and d= k csφ his can b win as d a 4 8 ad a I ε p Ip m π p= 8 Thus h ingal y is / d a y ik y y/ k ε y 4 8 εad a y m = p pm π p= 8 d J I I ip ipπ Singl Vlciy Cmpnn Elcsaic Wih h ingals ach h indiidual lciy cmpnns cmpld h suls can b subsiud back in quain 35 g as a uncin a sing lciy cmpnn. Sinc LIF is ypically dn in h ppndicula paalll dicin lai h backgund magnic ild his suggss ha shuld b a uncin and y. As and y nly n dicin is ndd bcaus y as a uncin n h h is quialn du h symmy h paicl min. Sinc his min is symmic alng h ais h cdina sysm can b ad aund h ais making y suicin ppndicula masumns. Blw is y quain 9 h sam as Saay al. 7 Φ ik ω T T y = y Z ζn m ζ ζn m π h nm T T k y / imπ imn θ ik ε y a /8 c /4 Jm p Wih h chang sumain indics in quain 68 aid cnusin ms quain 9 is. ε m n k / Φ ik ω imπ i m n φ = π mn ε n m k h ω d 4 εad a ipπ Ip Im p p= 8 π c /4 εac a Il In l il l= 8 a εac I n p/ Ip 8 ipπ / 9 9 5

19 . Cnsain w k^ and k using h Dispsin Rlainship As wih any wa h is a dispsin lainship dins h lainship bwn h quncy w and h wa numb k. In his cas lcsaic plasma was ha bn assumd s h dispsin lain lcsaic was is usd quain 9. This quain las ω k and k pids an addiinal cnsain. This allws h liminain anh aiabl m quain 9 9 and impss h wa lainship ha pubs h lciy disibuin. Slcing h pubing wa is h ms impan pa h cnsain bcaus i is h wa lngh his wa ha h masumn is amping dmin. Thus his inmain shuld b cnaind in h pcss smwh. S lcsaic was h dispsin lain is a k ε k k s ω = k k kds In as i n nt Z ζn ζn = k h T 9 wh k ds is h dby wa numb h sum i and a lcns and ins. All h paams a h sam as b. Using all his inmain h hical al and imaginay pas y can b gnad lcsaic in cycln was. Th hical al and imaginay pa w din y s using quains 9 and 9 a shwn in Figu and Figu din ss plasma paams. On impan hing n h is ha h a m ha causs h phas as a uncin lciy. I is h pik y / m. 6

20 .8 Signal Ampliud Ab Imaginay al y / Figu : Ral and Imaginay pas an y an agn plasma cnind by a 4 gauss magnic ild wih appimaly a cm/s di in h dicin and an ispic mpau. V. Th wa numbs h in cylcn wa a k.5 cm - and k.58 cm -. 5 Signal Ampliud Ab al imaginay y / Figu : Ral and Imaginay pas an y an agn plasma wih a gauss magnic ild n di in h dicin and an anispic mpau T /T =. Th wa numbs h in cycln wa a k 3.8 cm - and k.6 cm -. 7

21 3 Elcagnic Th masumn can b gnalid includ lcmagnic was as wll. This mans a hical calculain a ull lcmagnic nd b cmpld. T d his Faaday s law can b usd la h lcic ild h magnic ild. B E = Again using aling was h magnic ild B can b win in ms h lcic ild E. ik E = iωb k E B = ω Subsiuing his in quain 8 m scin. h lcmagnic is q k E = d [ E ] ω Subsiuing in h aling wa sluin in quain 96 pducs q [ k E ] ik ω = d E ω Nw using c idniis quain 97 can b win as q = d [ E Ek k E ] ω ik ω 98 As b a chang aiabl is mad -. Using τ= - and h lainships h psiin and lciis in scin. quain 98 can b cnd a simpl ingal τ. T d his sa by wking wih h m in h squa back and cmpling h c pains. Thn quain 98 h m in h back bcms 8

22 [ E Ek k E ] = ω E E E y y E k k k y y k E ˆ ω ω Ey E E y y E ky k k y y k Ey yˆ ω ω E E E y y E k k k y y k E ω ω ˆ ˆ yˆ ˆ 99 [ E Ek k E ] = ω E E E y y E k k k y y k E ω ω E y y E E y y E k y y k k y y k E y y ω ω E E E y y E k ω k k y y k E ω Wking hugh h algba and cllcing ms ach lcic ild cmpnn yilds [ E Ek k E ] = ω E k k ω ω E y y k k ω ω k y y k k y y k E ω ω ω ω 9

23 F simpliciy bh h lcic ild and h wa c a cnd cylindical cdinas. In ding s h sul applis nly lcmagnic was ha a cylindically symmic i.. ciculaly plaid was. F linaly plaid was his ansmain ds n wk and h calculain wuld ha b caid u m gnally. Using E E k k y y = E = E = k = k csφ sin φ csφ sin φ and k =k quain can b win as [ E Ek k E ] = ω E cs E sin φy k k φ ω ω k sin y cs k sin φ k φ φ y kcsφ E ω ω ω ω [ E Ek k E ] = ω E k k csφ sinφy ω ω E k sin cs k sin cs φ φ φ φ y y ω ω 3 4 Only ms cnaining and y a h m sin φ y cs φ Using h lainship and y m scin. his can b win as cs τ sin τ csφ sin τ cs τ sin φ y csφ = sin φ y y 5 6 Wih h hlp sm ignmy quain 6 bcms sinφ csφ = cs φ ε τ cs φ ε τ π y y 7

24 Using quain 7 = and / = T /T [ E Ek k E ] = ω A cs φ ε τ y cs φ ε τ π B wh k T k T A E Ek = ω T ω ω T ω 8 9 and B = E T T Subsiuing quain 9 and quain 8 in quain 98 and changing h ingain aiabl yilds q = τ cs φ ε τ y cs φ ε τ π d A B mn = Raanging his quain cnninc i bcms εk i ycs sin k ik φ φ ω q k y = dτ J J m n εk i ycs sin k i k y k ω φ φ J J m n in m φ imπ i ω ε n m k τ mn = A cs φ ε τ ycs φ ε τ π B in m φ imπ i ω ε n m k τ Th n h sps hlp duc quain in a simpl ingal τ. Ths sps pand h csin ms in pnnial m and pand h Bssl uncins. εk i cs sin y φ φ ik q k y k ω = dτ J J m n ] mn = A i i Ay i π i π φ ε τ φ ε τ φ ε τ φ ε τ B i ωε n m k τ inm φ imπ 3

25 k ik y k dτ J J m n εk i ω ycsφ sin φ q = mn = i n m k in m i n m k in m A Ay imπ ωε τ φ ωε τ φ i ωε nm k τ inm φ im π i ωε nm k inm im π τ φ i ωε nm k τ inm φ imπ B d εk i ycsφ sin φ i k ω q = τ mn = A k y k k Jm Jn Jn Ay k k y k y Jn Jm Jm k y k BJ J m n i ω ε n m k τ i n m φ imπ i ω ε n m k τ i n m φ imπ i ω ε n m k τ i n m φ imπ 4 5 Using h Bssl idniy m quain 3 quain 5 bcms εk i ik y csφ sin φ ω q = dτ mn = A k y n k k A k y m y J m J n J J n m k k y k y k i nm k inm BJ J ωε τ φimπ m n k ik y k J J m n εk i ω ycsφ sin φ q = mn = π i ω ε n m k τ n m in m φ im A B d τ k 6 7

26 Nw ding h ingal τ quain 7 bcms k ik y k J J m n εk i ω ycsφ sin φ q = mn = inm φ imπ n m i A k B ω ε n m k Subsiuing back in A and B quain 8 i bcms k ik ω q k y imπ in m J J φ = m n mn = k T k T n m E Ek ω T ω ω T ω k T i E T ε n m k ω Raanging his quain simpliciy gis εk i y cs sin k ik φ φ q y k ω = J J m n mn = imπ T n m in mφ E Ek Ek T ω ω k T i E T ε n m k ω 8 9 Equain is h is d pubd lciy disibuin uncin a gnal ciculaly plaid lcmagnic wa. Lking a his quain dincs wih h lcsaic can b sn. Equain has w pas an lcsaic pa and an lcmagnic pa. Th lcmagnic pa is h cul m k E -k E and h lcsaic ms a h indiidual E and E ms. I w l h cul m g E = k E /k quain ducs quain 35 an lcsaic wa. Thus ingain h lciis can yild h pubd disibuin as a uncin n cmpnn lciy. 3

27 3. Ingaing lciy cmpnns Equain can b spaad as quain 35 cllc h ms wih ach cmpnn lciy gh. Th and y ms a h sam as in quain 35 s h suls h ingain hs ms m h lcsaic cas can b usd h lcmagnic cas. Th nly lciy cmpnn ha is din m h lcsaic cas is as can b sn in quain. In his scin h ingain is dn. 3.. Ingain Th ingal is i ik ω imπ / i m n φ = mn π π / y i ky/ ε Jm k y / i ky/ k ε J n / T π E ke ke T ω ω n m T E k T ε n m k ω Using h las m in quain h ingal is / T d E ke ke T π ω ω This ingal is cmpld by is making h llwing subsiuin s = d = ds Wih his subsiuin quain bcms n m T E k T ε n m k ω 3 4

28 / s E ke ke T n m T ds s s E s π ω T k T k s δm n wh δ mn is as b ω ε n m k δ n m= k Th ingal in quain 4 can b cmpld using h plasma dispsin lainships in quains 4 and 43. Th sul h ingain is Using h llwing plasma dispsin uncin lainship quain 6 can b win as Thus h ingal is E ke k E T Z δm n Z δm n Z δm n k ω T Z δ = [ δz δ] n m T E Z δm n k T E ke ke T Z δm n δm nz δm n Z δm n k ω T n m T / T d E ke ke π T ω ω E δm nz δm n k T n m T E = k T ε n m k ω E ke ke T T Z δm n δm n Z δm n k ω T T n m T E δm nz δm n k T 5

29 3.. Singl Vlciy Cmpnn Elcmagnic Using h ingal in his scin quain 68 and h ingals and y m scins.. and..3 a pubd lciy disibuin uncin an lcmagnic wa can b win as a uncin nly y s discussin in scin..4 and. Ths quains a ik ω E ke k E y = y Z δ m n π nm k ω T T δm n Z δm n T T n m E δm nz δm n k T J ik ω = mn m T k y imπ i m n θ ik ε / /8 y a c /4 a ε ac I n p/ Ip 8 i p imπ / i m n φ d 4 εad a ip π I Im p p 8 p= π /4 ac a il c ε I I n l l l = 8 ipπ / 3 3 T E ke ke T ω ω n m T E k T ε n m k ω wh all symbls ha bn dind in pius scins. 6

30 4 Las Inducd Flscnc masumn Nw ha h hical calculain h pubd disibuin has bn cmpld h chniqu masuing h pubd disibuin is dscibd. This scin will discuss h masumn. Th a h pssibl chniqus ha can b usd masu. Th ms cmmn and h simpls mhd uss a lckin amplii. This alng wih h basics masuing will b discussd in scin 4.. Scin 4. will discuss using a digii masu and h las scin 4.3 will discuss h pssibiliis using a css pw spcum chniqu h masumn. 4. asumn wih a Lck-in amplii B discussing masumns a bi discussin masumns is ndd. Typical LIF masumns in agn 39 pump h masabl sa a λ 6.5 nm and cllc h mid ligh a λ 46. nm as shwn in h ll diagam Figu 3. 4p F 7/ 46 nm missin 6.5 nm pump las 4s D 5 3d G 9/ Figu 3: Cmmnly usd LIF agn schmaic. By scanning h las a naw quncy band ha includs h λ 6.5 nm abspin lin whil masuing h innsiy h mid ligh h lin shap can b masud. Th shap his lin is dmind by sal lin badning mchanisms 9 bu in mpaus ga han. V and magnic ilds lss h. kg Dppl badning dminas. This allws a dic clain bwn h in lciy disibuin uncin and shap h cllcd ligh m h las scan. Th pblm backgund ligh a λ 46. nm is dad by chpping h las ligh and using a lckin amplii masu pumpd missin a h chpping quncy. By nly masuing h ampliud h missin a h chpping quncy h lck-in amplii incass h signal nis ai. A ypical schmaic h appaaus is shwn in Figu 4. 7

31 Las Singl quncy md Lck-in Chpp B PT Plasma Oupu Signal Figu 4: Typical LIF schmaic masuing h h d disibuin uncin. A ypical LIF masumn mad in h H helicn pimn HELIX is shwn in Figu 5. H h ais is h las quncy and h widh h lin is du Dppl badning. A i h ligh innsiy cu using quain 3 can b usd dmin h mpau h ins alng h ais h las s dicin. I I T υ = υp.779 υ υ / agn 3 H I is h pak innsiy ν is las quncy ν is las cn quncy T agn is h in mpau and h cicin in h pnnial.779 has bn calculad spciically agn..7.6 LIF Signal ab unis Fquncy Shi m Cn Agn In Lin GH Figu 5: A ypical LIF ac in Hli wh h blu lin is h signal and h d ds a h cu i. This in lciy disibuin has a mpau. V. 8

32 Wha is din abu h masumn is d pubain h lciy disibuin uncin? Th masumn h h d lciy disibuin uncin nly cas abu h innsiy ligh a h chppd quncy as a uncin lciy. Th masumn h is d pubd lciy disibuin uncin is cncnd wih h innsiy ligh lucuaing a h quncy h pubing wa. As a paicl llws h pah is gy-min is lciy is bing acd by h wa. Th wa is spding i up and slwing i dwn a h quncy h wa. Ths scillains in h paicls lciis chang h numb paicls a ach paicula lciy in a pidic mann i.. pubains h lciy disibuin uncin. Ths pubains a dicly ppinal h lucuain in h innsiy ligh any id lciy las s psiin. Th innsiy h lucuains pubain a a paicula lciy lls hw sngly h paicls wih ha lciy cupl h wa. Scanning h las h lciy disibuin uncin h innsiy h pubain as a uncin lciy is masud. Th innsiy alng wih h lai phas h scillain pubain is a masumn h is d pubd lciy disibuin uncin. A schmaic h pimnal sup is shwn in Figu 6. Nic ha a chpp is n usd his masumn bu h lckin nc signal is h signal ging h annna usd launch h wa. This allws h lck-in masu h pubd disibuin a h quncy h wa din in h plasma by h annna. F h lck-in uncin pply his h nc signal mus b a clan sinusidal squa wa. Las RF Singl Fquncy md Lck-in Annna B PT Plasma 46. nm il Oupu Signal Figu 6: A schmaic diagam h masumn h pubd lciy disibuin uncin. 9

33 Using his mhd y in cycln was gnad in HELIX by an addiinal haing annna ha bn masud 3. Th masud y alng wih h hical cus using h hy m sc. a shwn in Figu 7. Th i gis wa numbs k.5 cm - and k.44 cm -. Ths wa numbs a cnsisn wih lcsaic in cycln wa in a plasma wih a 4 V lcn mpau as in HELIX. a Singal Ampliud Ab y / h b..8 Singal Ampliud Ab y / h Figu 7: a shws h masud and hy cus smh lins h al pa h y. b shws h masud and hy cus h imaginay pa h y. 3

34 4. asumn Using Aagd Tim Sis An alna mhd using a lck-in amplii is us a digii cd h PT signal dicly. This mhd n nly pids inmain abu bu can pduc inmain abu and pubd disibuins a h quncis bsids h nc quncy. This inmain can n b baind dicly wih a singl lckin basd masumn. A lck-in amplii uss h nc signal gna a sin and csin signal a a singl nc quncy. Th lck-in aks ach hs signals muliplis hm wih h inpu signal and ingas a id lngh im ingain im n lck-in. Th sul is h Fui ampliud h inpu signal a h nc quncy. Th sin and csin pas gi h al and imaginay pas h Fui ampliuds which can asily b und in ampliud and phas. Wih a lck-in his is dn using pcisin analg lcnics. In his day and ag digial lcnics i is pssibl d his lcnically. Hw h pcisin lcnics in h lck-in amplii can pick u small signals i.. yild gd signal--nis ais small signals. Wih h digii his is dn by aaging mulipl im sis. Sinc pubd disibuin masumns a signals ha a phas chn wih a diing wa signals ha kp h sam phas a gin lciy im h digii can b iggd by a nc signal m h diing annna a h sam pin in im whil dicly cding h PT signal sal im sis. Ths im sis a aagd bing h signal u h nis sinc nly signals ha ha h sam phas in ach im sis will add whil andm signals will cancl u. This chniqu is simila bca aaging using a muli-channl aaging. F pubd lciy disibuin masumns an aagd im sis is akn sal din lciis las s quncis sling h lciy dpndnc h disibuins. An ampl his masumn is shwn blw in Figu 8. Th cls in h pl psn h innsiy h im aagd PT signal as a uncin im and lciy las quncy Tim s Las s psiin GH Figu 8: Innsiy pl h aagd im sis as a uncin lciy las quncy. 3

35 Figu 8 claly shws h PT signal is pidic. This pidiciy cspnds h diing quncy and i is cla ha appimaly h wa pids a cdd alng h im ais. Lking acss h las s quncy ais i is cla ha h signal innsiy has sm lciy dpndnc. This is idnc ha h is a chn LIF signal in his daa a las a h diing quncy. N ha n h las s quncy ais h is a pidic PT signal h las ga han 4 GH and lss han 4 GH. Th las s quncy is usid h Dppl badnd gin wh h is n LIF signal bu h is spnanus missin a 46. nm m h ins in h plasma. Th min hs ins is acd by h wa making h mid ligh pidic in nau. This c is als sn wih h lck-in amplii bcaus h Lck-in signal ds n g usid h Dppl badnd gin. Th signal m h backgund ligh is subacd u in h analysis. T analy h daa h daa a cnd m h im dmain h quncy dmain sinc h pubd lciy disibuins a a uncin quncy. T pu h daa m Figu 8 in quncy spac h FFT ach aagd im sis is akn and pld agains lciy giing bh a pl h al and h imaginay pas h FFT. Th pubd lciy disibuin is h signal ampliud as a uncin lciy a singl quncy. I h a any signals ha a phas chn wih h nc digii igging signal a any quncy a cla signal can b sn in a pl h signal ampliud sus lciy. This is h adanag h lck-in mhd which can nly masu a h nc quncy. T gi a b illusain wh h pubd lciy disibuins a in h daa m Figu 8 h pw spcum ach aagd im sis is dn and h signal ampliud is pld sus quncy and lciy in Figu 9. Fquncy H Las s Fquncy GH Figu 9: Innsiy pl h pw spcum as uncin quncy and lciy las quncy. 3

36 Fm h innsiy pl in Figu 9 h a u cla bands ha ccu a uniqu quncis and all lciis. Ths a h quncis wh cla phas chn inmain is h quncis wih h lags Fui cmpnns. I is als cla m h qual spacing hs bands alng h quncy ais ha h signals a a hamnic quncis h diing quncy. Ths masumns w mad using 35.5 kh diing signal launch h was. Figu has gaphs h al and imaginay pas h FFTs gnad m h daa shwn in Figu 8 h quncis wih h ms signal in Figu 9. In ach h gaphs in Figu h backgund has bn subacd u m h FFT inmain Signal Ampliud Ab Signal Ampliud Ab Las Os Psiin GH Las Os Psiin GH.5 Signal Ampliud Ab Las Os Psiin GH Figu : FFT spcum inmain baind m h aaging mhd a h dc s b al d and imaginay blu a 35.5 kh and c al d and imaginay blu a 7 kh. 33

37 Th d band in Figu 9 cspnds h dc cmpnn h FFT and is shwn in Figu a. This is h h d lciy disibuin uncin h ins. Figu b shws h al and imaginay pas h y a h diing quncy. Figu c als shws a phas chn signal a h is hamnic h diing quncy. T undsand acly wha his pubd lciy disibuin is a hy is ndd. This may b a scnd d pubain h lciy disibuin du wa-wa cupling i may jus b y a wa cid a h hamnic quncy. S h hy a pubd lciy disibuin du wa-wa cupling is ndd cmpa i h hy h pubd lciy disibuin a h hamnic quncis dmin which yp pubain hy a. Th cmbinain hs h gaphs shws h pw his mhd masuing pubd lciy disibuins. Wih a singl masumn all h inmain ndd analy h was in h plasma is baind; h lciy disibuin uncin h is d pubd lciy disibuin uncin and any h phas chn pubains h lciy disibuin uncin. 4.3 Pssibiliis Using Csspw Spcum Tw mhds masuing pubd lciy disibuins ha bn discussd: using a lck-in and using digiid aagd im sis. Hw bh mhds qui a singl quncy nc signal idniy h pubd lciy disibuins. I may b pssibl masu h pubd lciy disibuins wihu a singl quncy nc signal by masuing h csspw spcum bwn a nc signal and a LIF PT signal. Nw as wih h aagd im sis h al and imaginay pas h csspw spcums culd b addd gh s ha quncy cmpnns h w signals ha a m sngly clad and ha lag Fui ampliuds will bcm lags a gin numb summd csspw spcums. Thn pling h al and imaginay ampliuds as a uncin lciy a singl quncy will gi h pubd lciy disibuins. This mhd will b plagud by h sam shcming h aagd im sis in h sns ha a lag numb indiidual csspw spcums will b quid incas h signal--nis ai. Hw his mhd shuld pid a masumn any phas chn inmain a any quncy as wll as shw inmain abu signals ha ha small lucuains ini widh in quncy. This mhd culd pn up pubd lciy disibuin masumns a wid ang was using LIF. Als i dinial ngy masumns can b mad n lcn disibuins hn his chniqu culd b applid pubd lciy disibuin masumns lcns. 34

38 Appndi A Th alab cds in his appndi a h ns ndd pduc y. This cd includs a di alng B and a mpau anispy wih spc B. Th plasma dispsin uncin is cusy DSua-achad and is nly alid al agumns. Funcin calculaing y : ************************************************************************ uncin [ans] = yplusakwhat % Win by Jhn Klin % Da: Apil % % Th cd is simila DSua-achads' y cd bu adds % mpau anispy and di alng h ambin magnic ild. % This cds als ds h summain in a din d h % Dsua-achads. % *This cd quis h uncin which is h plasma dispsin % uncin. % H h aiabls a % a = k_pp*h Dimnsinlss Kpp % k = Wci/K Vh Dimnsinlss /Kpaalll % W = W/Wci Dimnsinlss wa qncy % ha angl bwn and y dicin in plan pp % B % T = T/Tpp Tmpau ai % = Vdi/Vhpp Dimnsinlss di lciy ma = 4; % Numb ms sumains nm and p. phi =.7*^-; % Elcsaic pnial c=a.*sinha; Kpp*h*sinha d=a.*csha; Kpp*h*sinha % Dimnsinlss Ky aiabl qual % Dimnsinlss K aiabl qual k = k/sq; Wci/K*Vh*sq = /sq; i = sq-; % Dimnsinlss K aiabl qual % Dimnsinlss Di lciy % us imaginay numbs 35

39 % H is h sum n all m and p. n = ; % iniiali nwn n=-ma:ma m = ; p = ; % Sa lp p. H nly h ms wh n-p/ a ing % a alid. Thus w nly calcula a m i n-p/ has n % maind % i.. mduls n-p/ quals p = -ma:ma i mdn-p == % Ths a h mdiid Bssl uncin ms in y nwp=besselipa*c.*besselin-p/a^/4.*p-*p*pi/; ls nwp=.; nd p=pnwp; %sum ach h p ms nd %nd lp p %sa lp m m=-ma:ma a = W -nm*k-; %agumn plasma dispsin uncin a = W*k-; %agumn plasma dispsin uncin %wih n=m= % This m cnains all ms wih m's in hm alng wih h % plasma. % dispsin uncin and mpau anispy ms. nwm = a*a*t-t*a.*besseljma*....*p-.*.^./.*pi*d*....*p-i*m*pi/*pi*mn*ha; m = nwm.*p m; % sum ach h m ms nd %nd lp m n = nm; % sum ach h n ms nd % cicin h quain. c=.6.*.^-9.*phi./.99.*.^-3; %un alu: This has h cicin m and all ms ha % a n dpndn n h summains in y. ans = c.*n.*p-*a^/4-*c^/; 36

40 Plasma Dispsin Funcin: DSua-achad s uncin h plasma dispsin uncin. B caul h is a singulaiy a.. ********************************************************************** uncin ans=_in %This is h plasma dispsin uncin %calculad by cninud acins iimagng=imag_in <.; =_in.*~iimagng; =cnj_in.*iimagng; =; %imag > s w a k %imag < s b caul d= *i; d= *i; d3= *i; d4= *i; d5= *i; % n=-d; n=-d; n3= *i; n4= *i; n5= *i; % n=nssi; ans=ssi; m=:5 j=6-m; ans=nj*./nans*dj; nd ans=ans./; %his is h ld cd %ans=alansn*p-.^; %his is h m gnal cas: c h imag pa b psii alans=alans; imagans=absimagans; ans=alanssq-*imagans; %imag cd psii ans=ans; ans=ans.*~iimagng; sqpnn=-_in.*_in; lagsqpnn=-; i alsqpnn > 6. lagsqpnn=; nd 37

41 i lagsqpnn < sqp=p-_in.*_in; %nw dpnding n h sign imag d sm adjusmns ans=cnjans.*sq-*sqpi*sqp.*iimagng; ans=ansans; ls ans=cnjans.*sq-*sqpi*..*iimagng; ans=ansans; nd 38

42 Bibligaphy F. Ski and F. Andgg Phys. R. L R. A. Sn D. N. Hill and N. Rynn Phys. R. L D. N. Hill S. Fnaca and. G. Wickham R. Sci. Insum J. E. Dummnd Phys. R R. Z. Sagd and V. D. Shaan Pc. nd Inn. Cn. Gna N. Rsnbluh and N. Rsk Pc. nd Inn. Cn. Gna Saay S. D Sua-achad and F. Ski Phys. Plasmas D. G. Swansn Plasma Was Acdmic Pss Inc R. F. Biin WVU Innal Rp PL P. A. Ki E. E. Scim and.. Balky Phys. Plasmas E. E. Scim P. A. Ki. W. Zinl.. Balky J. L. Klin and. E. Kpk Plasma Suc Sci. Tchnl P. A. Ki Ph. D. Dissain Ws Viginia Unisiy J. L. Klin E. E. Scim P. A. Ki.. Balky and R. F. Biin Phys. Plasmas