LAC-P-637 Jauay 6 Aalyss of Las-Dv Patl Alato fom Plaa ft Codutv oudas * T. Pltt.L. Gzto Laboatos tafod vsty tafod CA 9435 Abstat Ths atl xplos th gy ga fo a sgl latvst lto fom a moohomat laly polazd pla wav dt o a plaa fltv bouday otd at a abtay oblqu agl ad ompas th pdto fo th gy ga fom vs Tasto adato mthod ad th lt fld path tgal mthod. t s foud that both mthods pdt th sam gy ga gadlss of th otato of th bouday. A bf aalyss o patally fltg sufas s pstd. * wok suppotd by Dpatmt of gy gat D-FG3-97476 Wok suppotd pat by Dpatmt of gy otat D-AC-76F55 LAC tafod vsty tafod CA 9439
LAC-P-637 Jauay 6. toduto Th qusto of las dv patl alato as a vs adato poss psts a tstg vw of th udlyg physs fo hagd patl alato. Th osvato of ltomagt gy statmt kow as Poytg s Thom [] s th t of th vs-adato ptu. t th as of stutu basd la patl alato th vs Tasto adato (T ptu pdts a gy ga qual to th ovlap tgal of las ad th patl s wak fld adato patts th fafld [3]. Hstoally th gy ga alulatos fo las-dv patl alato smop stutus typally utlzd th fld Path tgal Mthod (PM of th dt las fld o-popagatg wth th lto bam ad mad o assumptos about th alato stutu oth tha ts ablty to magally tmat th las fld [45]. At fst gla ths ass th qusto to th gal valdty of th path tgal gy ga mthod ad would pompt us to sk spal stas wh w would xpt T ad PM hav dfft gy ga pdtos. O suh addat stuato fo pottal dffs btw T ad PM am up th poof-of-ppl xpmt (th LAP xpmt [6] fo las-dv patl alato a sm-ft vauum. A lag flutuato th obsvd gy modulato was obsvd v at odtos wh th appad to b good spatal ad tmpoal ovlap btw th las ad th lto bam. As a possbl xplaato t was hypothszd that th gy ga flutuato was ausd by shot-to-shot vaatos of th otato th fltv bouday sultg ovlap vaatos btw th las ad th tasto adato flds. Th data show Fgu s a xampl of th flutuato of th gy modulato typally obsvd. 3 3 9 8 V (k 7 d a 6 y sp g 5 4 W HM 3 F FWHM gy spad (kv las o las off 8 85 9 95 3 35 3 35 3 las tm (ps las tmg (ps Fgu f th gy ga pdtd by th T poss dos dd dff fom th gy ga pdtd by th lt fld path tgal mthod fo boudas at oblqu otatos
LAC-P-637 Jauay 6 w would hav th possblty to tst ths wth ou xpmtal stup ad a futu st of xpmts at 63 w pla to masu th gy ga as a futo of a otolld oblquty agl of th fltv tap agg fom omal d to at last 45 dgs. Thfo a thoough aalyss of th pdtos fom th T ad th PM ptus po to th xpmt a of tst.. gy ga alulatos ad assumptos W hav to alulat th xptd gy ga wth a bouday at oblqu otato by both mthods. To do ths w wll mak th followg assumptos: Fo th sak of smplty assum that th las bam s moohomat ad that t a b dsbd by a pla wav. Fo th lag gaussa las bam w mploy th LAP xpmt ths s a vy good appoxmato. Ths dos ot ompoms th physs s w a dsb las bams of abtay spatal ad tmpoal pofls as supposto of moohomat pla wavs ad th aalyss fo both T ad PM s la. Th bouday s a pft hgh flto. Fo th mtall sufa of th bouday ad th N las ths s a faly good appoxmato. Absopto ffts wll b dalt a subsqut atl. Th lto tajtoy s staght ad s uafftd by th las o th bouday ad futhmo th lto s vloty vβ mas uhagd; a assumpto typally mad all path tgal gy ga alulatos. Th gy lost fom tasto adato a b gltd. W fst valuat th gy ga xptd fom th T ptu ad th pod wth th PM ptu.. gy ga fom th T ptu Wth th assumptos lstd al w a utlz Poytg s Thom ad fd that th gy ga of th hagd patl s d μ ( ds dt wh s th total lt fld ad s th total magt fld. Th total fld s th supposto of th las fld ad th tadd fld of th hagd patl. dots th path of th hagd patl ad s th sufa of th volum whh ompasss th tato. th fa fld th alatg - ad -ompots ( T ad T of th tadd flds domat. Ths a omal to ad omal to ah oth that s T T. Th las fld a b dsbd by a dt ompot wth flux tg ad a fltd ompot lavg th sufa. fgu. 3
LAC-P-637 Jauay 6 wak fld T T wak fld T T stutu lto path las dt fld wak fld T T las fltd fld Fgu sg th Fou tasfomato pa that has th fom ~ ~ t d t ( ( t dt ( t ( π ad th lato btw th dt th fltd las flds ad th tadd flds w a xpss th gy ga of th patl of quato as πz * ( ( T( dsd 3 Z wh s th vauum mpda. A dtald dvato of quato 3 s gv th appdx. As s quato 3 oly th fltd las fld ompot otbuts to th ovlap tgal th T ptu. Now w d to fd T(. Th tasto adato fom a flat ft fltv bouday a b ovtly foud by th mthod of mag hags. W assum a hag wth vloty β (omalzd to s q 4
LAC-P-637 Jauay 6 dt o a flat bouday at a oblqu agl. y th mthod of mag hags th s a ospodg mag hag q wth vloty β bhd th bouday. Ths tst at th bouday ad a b thought of omg to a sudd stop fftvly alg ah oth. T fom mag hag obsvato pot O φ ( ξ T fom hag hag q β β mag hag q q bouday Fgu 3 tatg fom th tadd sala ad vto pottals fo a pot hag A Φ t t ( x t ( x t μq 4π q 4πε v V ( t t ( x t; t ( t t ( x t; t dt dt 4 ad usg th lt ad magt fld xpssos d A Φ A 5 dt Th tadd lt fld of a alatg hag a b foud to b ad ( β qz d 4πK dt K 6 5
LAC-P-637 Jauay 6 th fquy sptum ths ospods to qz d β ( t ( x 4 dt π dt K 7 Assum th dlato tm fo th hag q s ftly shot h th tm ( t s appoxmatly ostat ov th tgato tm wh β hags valu. qz d β qz β ( x ~ dt π dt K ~ 4 4π K fal 8 usg β ad β β z w gt fal tal β β tal β T os osφ q Z β s 4π β os s ( φ os sφ 9 ot that th lt fld s adally polazd os osφ ( β os sφ s Fo th mag hag q w pfom xatly th sam typ of alulato xpt that t s ovt to todu a spaat otatd st of oodats algd wth th tal tajtoy of ths hag z // β. T os osφ q Z β s 4π β os s ( φ os sφ t s asy to vfy that th atsa oodats ( x y z a latd to ( x y by th otato matx wth a otato agl of π ξ about ŷ. z x xosξ zsξ y y z z osξ xsξ ad 6
x x y y z z ta x ta y x y z taφ y x z taφ y x LAC-P-637 Jauay 6 Th total fld s T T T. ou as w a tstd th T fld patt th xz pla (y. Ths allows us to xpss th fld patt as a futo of. sg q w obta q 3 T qz 4π os β s β s β os β os s 4 wh ( π ξ oluto th latvst lmt th latvst lmt th ovlap btw th T os of th hag ad th mag hag a glgbly small. th T o of th hag ls th fowad dto (bhd th bouday th las has vtually o ovlap ad w a appoxmat th T fld th vauum spa as th otbuto fom th mag hag os ( qz β s ( T 4π β os s ( Nxt w d to xpss th fltd las fld tms of th ( oodats. As show Fgu 4 assum that th dt las bam s a moohomat pla wav at a shallow agl α wth spt to th lto bam. H th las fld fltd fom th s f ( x y z s popagatg at a shallow agl to th z axs. ths sta th fa-fld patt ( of th las bam a b xpssd tms of th Fauhof dffato of th las fld at z. 5 f k ( x y f ( u v ux vy k k k dudv π 6 Th ( u v pla s othogoal to th z axs ad gos though th og z. Not that k ad a b postv o gatv. o wh valuatg th tgal of quato 6 w hav to wath th sg of spally wh w tak th absolut valu of k ad xpss t tms of th wavlgth λ. f th bouday s a flat pft flto th fltd las bam s also a pla wav. Assum that th pla wav has th fom ( t P ( k os t wth 7
LAC-P-637 Jauay 6 k x k α k k ad. Th offt k x α dats that k y z k x k th pla wav s at a agl α wth spt to th z axs. P vto of th pla wav. k s th polazato ut z x fltd las bam α x dt las bam lto ξ ( uv bam α ξ Fgu 4 ẑ th fquy doma th ampltud of th lt fld s ( ( ( ( k αu kzz ( kαu kzz u v z π ( 7 Thfo th fa-fld dffato ampltud s f ( x y ( ( k αu ( kαu ( k x y πk ku kv dudv 8 π (Th quatty bakts s ot a vto; t s a sum. Not that wh k k ad wh k k that s k a b postv o gatv. quato 8 w hav th tgal of a omplx xpotal whh gvs a dlta futo: f k ( a b x dx ( π k ( a b ( x y 9 x ( ( ( k k α π y ( k x ( ( ( k α 8
LAC-P-637 Jauay 6 wh ( fo > ad ( fo <. ( ( ( ( k x x y y x f α α λπ whh smplfs to ( ( ( ( ( k k x y y x f α πλ th small agl appoxmato th oodats φ a latd to y x by y x y x φ ta 3 s ( ( ( ( φ s a v a u (swthg to sphal oodats ( ( ( ( ( ( ( k k f α φ π πλ φ s 4 Th fltd las lt fld vto s ( ( ( ( ( ( ( ( k k P P f s α φ π πλ φ φ 5 Wth th ovlap tgal of th T fld ad th fltd las bam w a valuat th gy ga Δ ( ( ( φ π d d d Z T Δ s * ( ( ( ( ( ( ( ( Ω φ β α φ π λ π π π d d P P d qz Z T k k s os s 4 6 9
LAC-P-637 Jauay 6 at y φ π ( ( os os os ρ P T P s ρ os ρ. k s ( s ( os ρ β α ( s os ρ qλ 7 π β osα wh χ s a latv phas agl of th tasto adato at fquy. qλ sα β os ρ s 8 π β osα th small agl appoxmato w a us s α ~ α latvst lmt w a appoxmat β ~ γ. Thfo os α ~ α ad th qλ α Δ os ρ s 9 π α γ As xptd s popotoal to ad λ dpds o th polazato agl ρ ad Δ o th optal phas ad shows th xptd dpd o th las ossg agl F α α α γ that has a maxmum at α γ. ( ( max ± V. gy ga fom th path tgal mthod latvst lmt Fo a pla wav at a agl α wth spt to th lto bam dsb th lt fld of th dt las bam by ( t P ( k t os wh k x k α k y kz k k x ad k. Allowg fo th polazato agl ρ th polazato vto of th las fld s osα os ρ P s ρ 3 sα os ρ Alog th lto bam tajtoy ( z t th lt fld s 3
( z t P os( kzz t P os( k osα z t LAC-P-637 Jauay 6 3 Assumg ostat vloty t a b lmatd by t z Th gy ga s ( z t P os( ( k osα β z P os( ( osα β kz β. Thfo 33 q d q ( zdz 34 Fo ow assum that lto bam s latvst ad h th slppag dsta fo th fltd las bam s vy small ompad to th slppag dsta of th dt las bam ad thfo otbuto to fom th fltd las bam s small ompad to th otbuto fom th dt las bam. Th q zdz q os (( osα β kz P z dz 35 P z sα os ρ ~ α os ρ fo small agls α ad thfo os (( osα β qα os ρ kz dz 36 Th tgal s of th fom uv ( u φ du lm os( u φ du sφ os v 37 Thfo th gy ga of th hagd patl s qα os ρ λ π α γ qλ α os ρ s π α γ os ( u du αq os ρ λ π s α γ 38
LAC-P-637 Jauay 6 Not that s π fom th Fsl odto of flto fom a mtall sufa quato 9 ad quato 38 a dtal llustatg th quval btw th T ad th PM ptus at last th hghly latvst lmt. o what about th gy ga pdtos of ths two mthods th low gy lmt? Th assumptos lstd th toduto do ot mak ay statmt about th patl s vlots ad h w should xpt th T ad th PM mthods to gv dtal gy ga pdtos. V. Gal oluto ludg th low gy lmt th low gy lmt th pvous assumptos fo hgh γ bak dow. th PM ptu w aot glt th gy ga otbuto fom th out popagatg las bam ad th T ptu w hav to lud th tadd flds of both hag ad mag hag. Th T ptu Th T flds fo th hag ad mag hag stat to hav sgfat ovlap ad th otbuto fom both has to b ludd fo th total tasto adato fld quato 4. W had foud that th tasto adato patt had th fom T ( qz 4π β s β os os ( π ξ β s ( π ξ β os s 39 Th two tms th bakts pst th agula dstbuto futos of th tadd flds of th hag ad mag hag. At low γ th domatos of ths two tms do ot appoah zo at a patula agl ad h th adato patts do ot fom shap ad lag-valu paks but stat to hav sgfat ovlap at all agls. Ths sults a mo pooud asymmty th obsvd tasto adato os Fgu 5 shows a pola plot of th tsty of th agula dstbuto of th tasto adato fld of T at fou dfft bam gs:. ad 5 MV. quato 39 ( max
LAC-P-637 Jauay 6 45 9 6.8 9.8 6 5.6.4 3 5.6.4 3.. 8 8 33 33 4 7 3 β.5 MV MV 5 MV 4 7 3 Fgu 5; pola plots of T ( max fo a mtal fol at omal d ( ad a mtal fol at 45 t a b s that at th low gs th T dstbuto has a wd agula spad ad h th o asymmty fo th oblqu agl sufas s mo pooud. H fo fltv sufas at oblqu agls w xpt a asymmty of th gy ga dpdg wth whh sd of th T o th fltd las bam ovlaps. quato 39 dsbs th T patt. To alulat th ovlap tgal w d to xpss th fa-fld las patt gv tms of x y quato tms of. ( φ πλ ( ( ( ( k π ξ α φ ( ( ( P k s 4 wh P ( os os ρ s ρ s os ρ H th gy ga fom th ovlap tgal s πz πz Ω * ( ( T ( sddφd πλ β s β os qz 4π ( π ξ ( π ξ ( ( ( β s β os ( π ξ φ s d P P T sdω 3
λq s β sα β s π β osα β os LAC-P-637 Jauay 6 os os os ρ ( π ξ α s ρ ( π ξ α s s os ρ λq s β sα β s π β osα β os ( ξ α ( ξ α os ρ λq s sα s π β osα β ( ξ α os( ξ α os ρ wll wt ths xpsso as s( ξ α ( ξ α λq s os ρ sα Δ A F( α ξ; β π osα β os β 4 Wh F ( α ξ; β s th agula dpd futo of th gy ga. Th PM ptu Ca th PM mthod aout fo th sam pdtd asymmty of th gy ga as a futo of las ossg agl? As statd al at low γ th assumpto that Z >> Z o log holds: th slppag dsta of th fowad gog las bam Z dus lgth to a fw λ ad th otbuto fom th fltd las bam boms sgfat ad ths lmt both dt ad fltd las bam hav to b tak to aout. Th lt flds fom th dt ad th fltd las bam w dsbd by ( t P os( k t t P os k t ( ( wh P ad P dat th polazato stats. Now w d to fd th total logtudal lt fld at th oodats of th patl z. z z z 4 ( z z ( z t P os( k z t z z ( z t P z os( k zz t 43 P z P z k z k z th quatts ad a b foud fom th fgu blow 4
LAC-P-637 Jauay 6 fltd las bam z P z ξ α ξ α α k z lto bam ξ α ξ α k z ẑ dt las bam α P z P k k z P z z z sα os ρ s k osα k os ( α ξ ( α ξ os ρ Fgu 6 44 h th gy ga fom th dt ad fltd bam s q q s sα os ρ os ( k osαz t ( ξ α os ρ os( k os( ξ α z t dz dz 45 π (flto fom a mtall bouday ad t z β q q s sα os ρ os ( kz[ osα β ] ( ξ α os ρ os( kz[ os( ξ α β ] π dz dz 46 sg th fomula of quato 37 to valuat th os-tgal th valus of of quato 46 bom ad 5
LAC-P-637 Jauay 6 q sα os ρ k λq sα os ρ s π q s q s k ( osα β ( osα β ( u ( ξ α os ρ k( os( ξ α β ( ξ α os ρ s ( os( ξ α β os du os ( u du 47 H th total gy ga s Δ s( ξ α ( ξ α λq os ρ s sα Δ A F( α ξ; β π osα β os β 48 wh A ad F ( α ξ; β hav th sam valus as th gy ga alulatd by th T mthod show quato 4. Claly fo sufas at oblqu otatos ξ th gy ga Δ ( α ( α s ot symmt wth th las-ossg agl. bouday agl ξ/ bouday agl ξ/ 45 omalzd gy ga.8.6.4. β.5 MV MV 5 MV omalzd gy ga.8.6.4. β.5 MV MV 5 MV -3 - - 3 las ossg agl α (dgs -3 - - 3 las ossg agl α (dgs Fgu 7: Th agula dstbuto futo F ( α ξ; β F( α ξ; β max omalzd to ts maxmum valu fo dfft bam gs Fgu 7 llustats th omalzd gy ga futo fo th bouday at omal d ad at 45. As xptd fo th bouday otd at 45 t a b obsvd th s a la asymmty of th gy ga that s mo pooud at th low lto gs ad futhmo th s a ozo gy ga fo a dt las ossg agl of α. Ths s du to th lt fld of th fltd las bam bg paalll to th lto tajtoy ad dog wok fo a sub-wavlgth slppag dsta. Although t would appa fom Fgu 7 that ths fft s stogst at low gs t s 6
LAC-P-637 Jauay 6 atually lagst fo th hghly latvst as s at that lmt th slppag dsta appoahs F α ξ; β vsus th las-ossg agl α show fgu λ. A plot of ( 8 llustats ths. Th st shows ( α ξ; β α th futo ( α ξ; β F th vty of α ad shows that at F has th sam small but ozo valu fo th hgh patl gs. Oly th sta fo β. 5 shows a sgfatly low valu at α du to th dud slppag dsta at low lto spds.. 9 8 7 β.5 MV MV 5 MV.8.6.4. F(αξ;β F(αξ;β 6 5 4 3-3 - - α (dg -3-3.5-3 -.5 - -.5 - -.5.5 x -3 lto tajtoy - -8-6 -4-4 6 8 las ossg agl (dgs 45 sufa Fgu 8 Lt s xplo ths sta of havg th dt las bam at α to th as wh th hgh flto sufa s otatd fom omal d ξ to almost at gazg d fo th lto bam suh that th fltd las bam slppag dsta s sgfatly asd o th T ptu th fltd las bam s optmally ovlappd wth th T patt. gy ga agula dpd gy ga agula dpd omalzd F(αξ;β 9 8 7 6 5 4 3 β.5 MV MV 5 MV F(αξ;β / max( F(αξ;β.8.6.4. β.5 MV MV 5 MV 5 5 5 3 35 ξ bouday agl Fgu 9 5 5 5 3 35 ξ bouday agl 7
Fgu 9 shows th futo ( α ξ; β LAC-P-637 Jauay 6 F as th bouday agl s swpt fom to 8 ospodg to swpg th fltd las bam fom ξ to ξ 36. As xptd th optmum gy ga at hgh gs ous at vy shallow agls wh th fltd las bam s almost o popagatg wth th lto bam ad has a vy log slppag dsta (th optmum agl ow bg π ξ γ. At th low gs th maxmum gy ga s small (du to th dud slppag dsta but at th sam tm t s lss sltv to th fltd las bam agl ξ. t s tstg to ot that th gy ga fom swpg of th agl of th fltd las bam (th by tltg of th fltv bouday o by swpg of th put las bam agl s smply a pob of th tasto adato ampltud agula dstbuto. f th lto bam s optally buhd v th phas of th T patt ould b dtmd. V. Las alato th dowstam spa of th fltv bouday Th xampls volvd th aalyss of las alato th upstam spa of a hgh flto. Th aalyss th dowstam spa s ot muh dfft th th T o th PM ptu. Th T gy ga quato 3 s gal ad also appls ths sta ad th PM gy ga quato 34 s th sam xpt fo a hag of th path tgal lmts to q d q ( zdz 49 Th tasto adato patt a b foud th sam fasho as bfo; as th patl mgs to th dowstam spa th flds th dowstam spa a b aalyzd tms of a hag ad mag hag ovlappd ad at st abuptly movg at a vloty β to dfft dtos dpdg o th tlt agl of th sufa. p to a phas fato ths sults th sam typ of T fld patt as dsbd quato. Assum aga that th put ad fltd las fld s a pla wav ad th tlt agl of th sufa ξ s small. Th gy ga stll has th sam fom xpt that th dt las fld ad th fltd vsd ols. Now s o popagatg wth th lto ad shows a log slppag dsta whl s out popagatg to th lto bam ad h shows a vy shot slppag dsta. Th gy ga fo las alato th dowstam spa s s( ξ α ( ξ α λq os ρ s sα Δ A F ( α ξ; β 5 π os β osα β 8
LAC-P-637 Jauay 6 Th ma dff wth th as fo th alato upstam of th bouday s that ths tm th optmum ous at ξ α γ. Also swpg th tlt agl of th sufa o of th dt las bam pobs th ampltud of th dowstam tasto adato patt. V. Patally fltv bouday As statd thoughout th atl th bouday was assumd to b a flat pft flto. W may ask how quato 5 modfs fo a patally fltv sufa wth a flto offt <. To ga a fst asw w wll us th PM ptu to fd th gy ga. Wth a lss-tha-pft fltv bouday th ampltud of th fltd las bam boms ad thfo fo th upstam las alato w hav λq os ρ s sα s π osα β os ( ξ α ( ξ α β upstam 5 ad fo th dowstam las alato w gt λq os ρ s π os s( ξ α sα ( ξ α β osα β dowstam 5 th hghly latvst lmt th out popagatg tm boms glgbl ad w hav λq os ρ s sα upstam π osα β 53 λq ρ ( ξ α os s s dowstam π os( ξ α β Wth th PM ptu ad th hghly latvst lmt w xpt to s o sgfat fft fom a pooly fltv bouday fo las alato th upstam spa whl th dowstam spa w xpt Δ dowstam 9
LAC-P-637 Jauay 6 Δ upstam Δ dowstam upstam lto tajtoy dowstam < fltvty Fgu Th T ptu would pdt a dfft outom spally fo th upstam alato as whh fom th PM ptu appas to b dpdt of th bouday popts. Th T ptu would xpt a duto of th gy ga du to th dud fltd las fld ompot ad fom a dud T patt fom th lossy mdum. Th dspay ls th assumptos mad th T ptu wh th mdum s assumd to hav o ohm loss ad th oly ozo tms Poytg s thom a th lt fld path tgal of th tavlg patl ad th fa-fld adatv tms. Wth a lossy mdum Poytg s Thom would bom patl τ M J M dvdt M μ τ ( ds dt J M s th ut sd th mdum M. f th mdum s lossy th xtal lt fld ptats th mdum ad sd th mdum makg th volum tgal of th uts ad flds J M dv th mdum M quato 54 ozo. f th total volum quato 54 dd ot sto ltomagt gy dug th tast of th patl ad w would b lft wth M 54 patl τ M J M dvdt μ τ ( ds dt 55 H fo a lossy mdum th gy ga pdtd by th vs-adato ptu s ot ot. μ τ ( ds dt patl 56
LAC-P-637 Jauay 6 Th luso of a las-absobg bouday s a smpl xtso to th plad st of xpmts wth fltv boudas ad wll povd a xpmtal oppotuty to both tst fo th falu of th T ptu ad fo th valdty of th PM ptu ud a mo gal bouday odto. V. Colusos t has b show that fo pft fltv flat boudas th gy ga pdtd by th T ad th PM mthods yld xatly th sam valu gadlss of th patl s tal gy th otato of th bouday o th upstam o dowstam alato as. H w a ul out th hypothss that th obsvd gy flutuatos w du to adom tltg of th bouday sufa. A hgh flto bouday of th typ usd th pvous poof-of-ppl xpmt wth las alato aot dstgush btw T ad th PM ptus. Gv th pftflto assumpto ths fdg should ot b supsg at all s fo losslss mda Poytg s Thom ollapss to quato whh puts th path tgal gy ga ad th adato ovlap tgal o a qual footg. Howv th bf aalyss of th xptd patl alato th upstam spa of lossy boudas ( < appas to show a as wh th T ptu laly dffs ts gy ga pdtos fom th PM ptu. t s agud that th T ptu dos ot apply to ths stas baus of th appaa of a addtoal ohm loss tm of th matal that pvts Poytg s Thom fom ollapsg to th smpl xpsso of quato. Ths bf aalyss s a motvato fo a mo dtald tatmt of a mo galzd vs-adato ptu that luds lossy matals. Aoth tstg sta to b aalyzd s th as of a losslss taspat bouday of th appopat thkss that ould st th las phas ad fftvly doubl th gy ga. X. Appdx th fa fld go th lt ad magt flds hav th latos ad T T. Also th lt ad magt fld ompots a mutually othogoal ad oby T T A ( A
LAC-P-637 Jauay 6 T T T Not that th gatv tm fo th dt fld dats a wad flux of gy. Th total lt ad magt flds a T ad. H th tgal of quato ads T ( ( ( T T t ds ds d μ μ A3 whh xpads to ( ( ( T T T T T t ds ds ds d μ A4 Fo th vy fst sufa tgal A4 f th mdum s a pft flto th output fltd pow s qual to th put dt pow gadlss of th shap of th flto ad w hav ( ( ds ds A5 ad thfo th st sufa tgal dus to th wak fld adato tm ( ( T T T P ds ds Δ A6 Th sod sufa tgal of A4 a b valuatd fom th latos A. Th oss poduts hav th fom ( ( os s os s A7 Th sg vsal of th sod xpsso A7 wth spt to th fst s du to th oppost flux dtos of th dt ad fltd flds. Now w a s that th d sufa tgal of A4 dus to zo.
( ds LAC-P-637 Jauay 6 A8 Fo th 3 d sufa tgal A4 w a fd a smla st of valus fo th oss poduts T T T T T sφ ( T T sφ ( T T sφ ( T sφ ( T Not that fo th pa of T ad T th s o sg vsal s both fltd fld ad th wak fld hav a flux of gy th sam (outwad dto. H th 3 d sufa tgal ollapss to ( ds ( T T Thfo th total gy ga s T T A9 ds A μ ( T dsdt T A Takg to aout th assumptos lstd th toduto w a glt a lft wth T ad Z ( ( t T ( t dsdt A Wth th Fou tasfom dfto quato Z π ~ ~ ( ( ( ( t T dd dsdt A3 whh a b aagd to 3
Z π Z π ~ πz ~ ~ ~ ( ( T ~ ( t ( ( ds ~ T ( ( dsπ ( T dsd dt dd dd LAC-P-637 Jauay 6 A4 Fo al futos of tm T ( T( ~ ~ * ad thfo πz ~ ~ * ( T ( dsd A5 Ths povs th patl gy ga fomula show quato 3. X. fs. fo xampl Jakso Classal ltodyams d dto hapt 6 p 36. M. X A Fudamtal Thom o Patl Alato Podgs of th 3 Patl Alato Cof (3 3. Z. Huag G. tupakov ad M. Zolotov Calulato ad Optmzato of Las Alato Vauum Phys. v. pal Tops - Alatos ad amsvol. 7 3 (4 4. J.A. dghof.h. Patll gy xhag btw f ltos ad lght vauum Joual of Appld Physs 5 p 6-6 (979 5.. say P. pagl J. Kall Las Alato of ltos Vauum Physal vw 5 p 5443-5453 (995 6. T. Pltt.L. y. Colby. Cowa C.M.. as J.. p.h. ma Poof-of-ppl xpmt fo las-dv alato of latvst ltos a sm-ft vauum Phys. v. T Al. ams 8 3 (5 4