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.
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 4 3.. Ingain 4 3.. 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 3 4.3 Pssibiliis Using Csspw Spcum 34 APPENDIX A 35 BIBLIOGRAPHY 39 ii
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
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
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 =
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
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
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
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 7 8 9 3 6
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 33 34 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
. 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
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 38 39 4 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] 4 4 43 44 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 45 46 9
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
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 55 56 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 59 6 6
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 63 64 65 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 67 68..3 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
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 7 7 7 73 74 75 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 76 77 quain 74 bcms π / y k k i εcsφ sin α εcsφ sin α k εcsφ sin α d y = 3
Subsiuing back in quain 7 w g π π k ε csφ sin α imα dα 78 79 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 8 8 83 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 84 85 86 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
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 87 88 89..4 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
. 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
.8 Signal Ampliud Ab..6.4. -. -.4 Imaginay al -.6-8 -6-4 - 4 6 8 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. 5 5-5 al imaginay - -8-6 -4-4 6 8 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
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 93 94 95 96 97 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
[ 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
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
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
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
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
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
/ 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 4 5 6 7 8 9 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
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
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
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.5.4.3... -8. -6. -4. -... 4. 6. 8. 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
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
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.8.6.4 Singal Ampliud Ab.. -. -.4 -.6 -.8 -. -. -4-3 - - 3 4 y / h b..8 Singal Ampliud Ab..6.4. -. -.4 -.6 -.8-4 -3 - - 3 4 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
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...5-5.5-5 Tim s 3.75-5 5.-5 6.5-5 -4-4 Las s psiin GH.6.7.8.9.3.3 Figu 8: Innsiy pl h aagd im sis as a uncin lciy las quncy. 3
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 5 5 5 3-4 - 4 Las s Fquncy GH -3-3 5 7 8 Figu 9: Innsiy pl h pw spcum as uncin quncy and lciy las quncy. 3
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. 4 6 35 4 Signal Ampliud Ab. 3 5 5 Signal Ampliud Ab. - 5-4 -4-4 -6-4 - 4 Las Os Psiin GH Las Os Psiin GH.5 Signal Ampliud Ab. -.5 - -.5 - -.5-4 - 4 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
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
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
% 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
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=-.77453859*i; d=-.644746838*i; d3=-.57546*i; d4=-.435588634*i; d5=-.34788656*i; % n=-d; n=-d; n3=.98388745*i; n4=.7483339*i; n5=.4585547*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
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
Bibligaphy F. Ski and F. Andgg Phys. R. L. 59 896 987. R. A. Sn D. N. Hill and N. Rynn Phys. R. L. 37 833 98. 3 D. N. Hill S. Fnaca and. G. Wickham R. Sci. Insum. 54 39 983 4 J. E. Dummnd Phys. R. 5958. 5 R. Z. Sagd and V. D. Shaan Pc. nd Inn. Cn. Gna 3 8958. 6. N. Rsnbluh and N. Rsk Pc. nd Inn. Cn. Gna 3 44958. 7. Saay S. D Sua-achad and F. Ski Phys. Plasmas 3 436 996. 8 D. G. Swansn Plasma Was Acdmic Pss Inc. 989. 9 R. F. Biin WVU Innal Rp PL-39 998. P. A. Ki E. E. Scim and.. Balky Phys. Plasmas 4 74997 E. E. Scim P. A. Ki. W. Zinl.. Balky J. L. Klin and. E. Kpk Plasma Suc Sci. Tchnl. 7 86998. P. A. Ki Ph. D. Dissain Ws Viginia Unisiy 999. 3 J. L. Klin E. E. Scim P. A. Ki.. Balky and R. F. Biin Phys. Plasmas 6 4767999. 39