Supeluinal Nea-field Diple Eleanei Fields Willia D. Wale KTH-Visby Caéaan SE-6 7 Visby, Sweden Eail: bill@visby.h.se Pesened a: Inenainal Wshp Lenz Gup, CPT and Neuins Zaaeas, Mexi, June -6, 999 Induin The pupse f his pape is pesen aheaial evidene ha eleanei nea-field waves and wave ups, eneaed by an sillain elei diple, ppaae uh fase han he speed f lih as hey ae eneaed nea he sue, and edue he speed f lih a abu ne wavelenh f he sue. The speed a whih wave ups ppaae (up speed) is shwn be he speed a whih bh dulaed wave infain and wave eney densiy ppaae. Beause f he siilaiy f he venin paial diffeenial equains, w he ysial syses (anei sillain diple, and aviainal adiain sillain ass) ae ned have siila esuls. Analysis f elei diple. Geneal sluin Nueus exbs pesen sluins f he eleanei fields eneaed by an sillain elei diple,. One siple and elean sluin slves he inheneus send de supepenial wave equain (pp. - 6). The eleanei fields an hen be deived f he Hez ve (Z). Fiue. Seial dinan syse used in pble: z x PDE (supepenial wave equain): E H E y Z () Sluin: Z R Cs( ) e i Z i Sin( ) e Z () Definin equains: C xz C R C C i Sin( ) i e () Field alulains: E xc B C whee: H B ()
Resulan eleial and anei field pnens f an sillain elei diple E H Cs( ) Sin( ) i e i i E Sin( ) i i e () i e (6) I shuld be ned ha his sluin is nly valid f disanes () uh eae han he diple lenh (d ). In he ein nex he sue ( ~ d ) an n be deled as a sinusid: Sin. Insead i us be deled as a sinusid inside a dia dela funin: d Sin. The sluin his hype-nea-field pble an be alulaed usin he Lienad-Wieha penials,, 8.. Lines f elei fe analysis Tadiinally he elei lines f fe an be deeined f he elain ha a line eleen (ds) ssed wih he elei field is ze. The esulin paial diffeenial equain an hen be slved yieldin he lassial esul. Resulan Equain: CSind CSind (7) Sluin: Sin Cs Tan Cns (8) A nu pl f his sluin (Eq. 8) eveals he lassial adiain sillain elei diple field paen (Fi. ). Caeful exainain f he paen eveals ha he wavelenh f he eneaed fields ae lae in he neafield (a) and edue a nsan wavelenh afe he fields have ppaaed abu ne wavelenh f he sue (b). The speed f he fields (ase speed = ) nea he sue an hen be nluded ppaae fase han he speed f lih f he elain ha wave speed ( ) is equal he wavelenh ( ) uliplied by he fequeny (f), whih is nsan. f if Fiue. Maheaia aniain nu pl f abve sluin a > b Radial Field (E ) Maheaia de used eneae aniain a final b a b = Tansvese Field (E )! " " #" " " " "! " " " " $% &!" #'" #(" #)" #" #" #" #" #)" #(" #'" *+
The Maheaia de (Ve..) shwn abve eneaes pls f he ppaain elei field a diffeen islaed ens in ie. Muse liin any f he faes in Maheaia aniaes he pl, evealin ha he vals f nsan elei field enlae as hey ppaae away f he sue (laed a = ). I is als ineesin ne ha as he elei field lines ae eneaed, se f he elei lines f fe vey nea he diple (~ / wavelenh) appea ppaae nly a sh disane and hen evese and ppaae ba in he sue. Fiue. Maheaia aniain f nu pl f elei lines f fe nea sue Analysis f ase speed and up speed. Phase alulain The eneal f f he eleanei fields (ef Eq.,6) eneaed by a diple is: i Field x iy e (9) If he sue is deled as Cs, he esulan eneaed field is: Field Ma Cs Ma Cs () whee: Ma x y I shuld be ned ha he fula desibin he ase is dependan n he quadan f he plex ve. y y Tan x Cs x () x y x. Definiin and alulain f wave ase speed Sinal pee ppaaes wih speed equal ase speed ( ) Phase speed an be defined as he speed a whih a Sinusidal sinal wave psed f ne fequeny ppaaes. The ase speed ( ) f an sillain field f he f Sin( ), in whih (, ), an be deeined by sein he ase pa f he field ze, diffeeniain he esulan equain, and slvin f.
( ) () Diffeeniain wih espe yields: () Cbinin hese esuls and usin yields: (). Definiin and alulain f wave up speed Pea f dulain pa f sinal ppaaes wih speed equal up speed ( ) In se ysial syses he wave ase speed Mdulaed sinusidal is a funin f fequeny. In hese syses when waves psed f diffeen fequenies ppaae, he wave up (wave envelpe) ppaaes a a diffeen speed (up speed) han he individual waves. The up speed is als nwn be he speed a whih wave eney and wave infain ppaae 6 (pp.68-69), (p.). The up speed f an sillain field f he f Sin( ), in whih (, ), an be alulaed by nsidein w Fuie pnens f a wave up: Sin( ) Sin( ) Sin( ) Sin( ) () in whih:,,, The up speed ( ) an hen be deeined by sein he ase pa f he dulain pnen f he field ze, diffeeniain he esulan equain, and slvin f : ( ) Diffeeniain wih espe yields: Cbinin hese esuls and usin he elain (6) (7) yields: li sall (8)
. Radial elei field (E ) Applyin he abve ase and up speed elains (Eq., 8) he adial eleial field (E ) pnen (Eq. ) yields he fllwin esuls: Fiue y x (9) Tan () ( ) () ( ) ( ) ( ) () Fiue Fiue 6 E E E 9 de. Tansvese elei field (E ) Applyin he abve ase and up speed elains (Eq., 8) he ansvese eleial field (E ) pnen (Eq. ) yields he fllwin esuls: y x () Cs () () ( ) ( ) 6( ) 7( ) ( ) ( ) (6) 6 8 Fiue 7 E 8 de Fiue 8 Fiue 9 E - E C vs E -
.6 Tansvese anei field (H ) Applyin he abve ase and up speed elains (Eq., 8) he ansvese anei field (H ) pnen (Eq. 6) yields he fllwin esuls: y x Cs ( ) ( ) (7) (8) ( ) ( ) ( ) (9) () Fiue H 8 de Fiue Fiue H H Gaial evidene f supeluinal ase and up speed. Supeluinal nea-field ase speed f adial elei field T densae he supeluinal nea-field ase veliy f he lniudinal elei field, he alulaed ase and apliude funins an be inseed in a sine sinal and he field apliude an hen be pled in he nea field as a funin f spae () a seveal islaed ens in ie (), (Fi. ). A field ppaain a he speed f lih (shwn as a dashed line) is als inluded in he pl f efeene. The fllwin paaees ae used in he subsequen pls: wavelenh (), GHz sinal fequeny (f),.ns sinal peid (T). The fllwin Maheaia de (Fi. ) is used eneae hese pls: Fiue. Maheaia de used eneae pls! " " " #" "!,"!+%,"," " " 6
Fiue. E vs Spae Csinusidal Sinal E T T T T The lniudinal field (shwn as a slid line in he pl abve) is bseved ppaae away f he sue, whih is laed a =. As i ppaaes away f he sue, he sillain apliude deays apidly (/ ) nea he sue ( < ), and deeases e slwly (/ ) in he fafield ( > ) (ef Eq. ). A field ppaain a he speed f lih (shwn as a dashed line in he pl abve) is als inluded in he pl f efeene. Bh sinals sa ehe in ase. The lniudinal field is seen ppaae fase han he lih sinal iniially when i is eneaed a he sue. Afe ppaain abu ne wavelenh he lniudinal elei field is bseved slw dwn he speed f lih, esulin in a final elaive ase diffeene f 9 deees. In de see he effe e lealy he sinals an be pled wih he apliude pa f he funin se uniy (Fi. ). 7 Fiue. E (Nalised) vs Spae Csinusidal Sinal E ( ) T ( ) T ( ) 7 T ( T ) I is als insuive pl he sinals as a funin f ie () f seveal psiins () away f he sue (Fi 6). A he sue ( = ) bh sinals ae bseved be in ase. Fuhe away f he sue he lniudinal field sinal is bseved shif 9 deees, indiain ha i aives ealie in ie. The pls shwn belw ae nalized f laiy, bu i shuld be ned ha he sinals have he sae f even if he apliude pa f he funin wee inluded. The nly diffeene is he veial salin f he pl. F hese pls i an als be seen ha he lniudinal field ppaaes uh fase han he speed f lih nea he sue ( < ), and edues he speed f lih a abu ne wavelenh f he sue ( ), esulin in a final elaive ase diffeene f 9 deees beween he lniudinal field (shwn as a slid line), and he field ppaain a he speed f lih (shwn as dashed line). 7
Fiue 6. E (Nalised) vs Tie Csinusidal Sinal E. Supeluinal nea-field up speed f adial elei field T densae he supeluinal nea-field up ppaain speed f he lniudinal field, he alulaed ase and apliude funins an be inseed in he speal pnens f an apliude dulaed sine sinal, and he field apliude an hen be pled as a funin f spae () a seveal islaed ens in ie (), (Fi. 8). T densae his ehnique he up ppaain (shwn as a slid line) is paed he ase speed ppaain (shwn as a dashed line) f waves f he f: Cs. Ne ha he ase pnen () is independen f fequeny. This esul is nwn pdue up waves and ase waves ha bh ppaae a he speed f lih. This an be seen by usin (Eq. ): sine. The fllwin paaees ae used in. Usin (Eq. 8): he fllwin pls: Caie pa f sinal - wavelenh ( ), MHz sinal fequeny (f),.ns sinal peid (T), Mdulain pa f he sinal - wavelenh, MHz sinal fequeny (f),.ns sinal peid (T). Ne ha he ase elain f bh sinals ae he sae a diffeen islaed ens in ie and ha bh sinals ppaae away f he sue a he sae speed. The fllwin Maheaia de (Fi. 7) is used eneae hese pls: Fiue 7. Maheaia de was used eneae hese pls 7 -.. * * * - %+$ - -. *.. *.. *....!-". " " " " " " Fiue 8. Lih Phase (Csine Wave) and Gup (AM Sinal) vs Spae T T T 7 T 8
Plin he sinals as a funin f ie f seveal spaial psiins f he sue als shws ha up and ase sinals avel a he sae speed and eain in ase as hey ppaae. Fiue 9. Lih Phase (Csine Wave) and Gup (AM Sinal) vs Tie The supeluinal nea-field up ppaain speed f he lniudinal elei field an als be densaed in he sae way as in he abve exaple. The alulaed ase funin f he field an be inseed in in he speal pnens f an apliude dulaed sine sinal and he field apliude (shwn as a slid line in he pl belw) an hen be pled as a funin f spae () a seveal islaed ens in ie (), (Fi., ). An apliude dulaed field ppaain a he speed f lih (shwn as a dashed line) is als inluded in he pl f efeene (envelpe ppaaes a speed f lih). Ne ha f his efeene sinal bh he ase speed and he up speed ae equal he speed f lih (ef Fi. 8, 9). The fllwin aheaia de (Fi. ) is used eneae he pls belw. The sae sinal paaees used in he pevius exaple ae used in he alulain. Fiue. Maheaia de used eneae pls -.. * * * * * * - %+$ - -. *.. *.. * -. *.. *..*....!-" -" " " "!," " " Fiue. E (Nalized) vs Spae AM Sinal E T T T 7 T 9
Fiue. Z f E (Nalized) vs Spae AM Sinal E T T The abve pls shw an apliude dulaed lniudinal field up pae (shwn as a slid line) ppaain away f he sue, a = (up axia aed by veial aw). A ppaain speed f lih up wave (shwn as a dashed line) is als pvided f efeene. The up axia f he apliude dulaed lniudinal field is bseved ppaae he ih side f he pl befe he up axia f he speed f lih wave. These seies f pls lealy densae ha he lniudinal up wave ppaaes uh fase han he speed f lih nea he sue ( < ). Afe ppaain abu ne aie wavelenh f he sue ( ) he dulain pa f lniudinal field (envelpe f slid line) edues he speed f lih, esulin in a final elaive ase diffeene f 9 deees (elaive he aie sinal) beween he lniudinal field, and he field ppaain a he speed f lih. I is als vey insuive pl he field apliude f he apliude dulaed lniudinal wave (shwn as a slid line in pl belw) as a funin f ie (), f seveal psiins away f he sue (), (Fi., ). As befe, an apliude dulaed wave avelin wih lih speed is als pled f efeene (shwn as a dashed line). A he sue ( = ) bh sinals ae bseved be in ase. Fuhe away f he sue he dulaed lniudinal field sinal is bseved shif he lef, indiain ha he dulain pa f lniudinal field (envelpe) aives ealie in ie. The pls shwn belw ae nalized f laiy, bu i shuld be ned ha he sinals have he sae f even if he apliude pa f he funin wee inluded. The nly diffeene is he veial salin f he pl. F hese pls i an be seen ha he dulain pa f lniudinal field (envelpe f slid line) ppaaes uh fase han he dulaed lih speed sinal (envelpe f dashed line ppaaes a speed f lih) nea he sue ( < ). Afe ppaain abu ne aie wavelenh f he sue ( ) he dulain pa f lniudinal field (envelpe f slid line) edues he speed f lih, esulin in a final elaive ase diffeene f 9 deees (elaive he aie sinal) beween he lniudinal field, and he field ppaain a he speed f lih. Fiue. E (Nalized) vs Tie AM Sinal T 7 T E 7
Fiue. Z f E (Nalized) vs Tie AM Sinal Z E (Nalized) vs Tie Z E (Nalized) vs Tie Z E (Nalized) vs Tie Z E (Nalized) vs Tie E (x -8 s) (x -8 s) (x -8 s) (x -8 s) 7 The fis fae f he pl shws he w wave ups sain in ase a he sue ( = ). The fllwin faes shw ha as he w waves ppaae away f he sue, he up axia (aed by a veial aw) f he apliude dulaed lniudinal aives ealie in ie han he up axia f a lih speed apliude dulaed sinal, hus densain ha he up speed f an apliude dulaed lniudinal eleial field is uh fase han he speed f lih nea he sue ( < ), and edues he speed f lih afe i has ppaaed abu ne wavelenh f he sue ( ), esulin in a final ase diffeene f 9 deees. 7 Relain beween up speed and ppaain speed f infain and eney Seveal auhs have indiaed ha wave up speed (ppaain speed f wave envelpe) is als he speed a whih wave eney and wave infain ppaaes 6 (pp.68-69), (p.). One inuiive way undesand his is aheaially apliude dulae and dedulae a ppaain wave, heeby ansiin and deein infain. An apliude dulaed wave an be aheaially deled as fllws: AM Si Cs Cs () whee ( ) is he dulain fequeny, ( ) is he aie fequeny, and () is he index f dulain. Usin inei ideniies he AM sinal an be shwn be: AM Si Cs[ ] Cs[ ] Cs[ ] () Wave ppaain an hen be delled by insein he alulaed ase elain f he wave in he speal pnens f he dulaed sinal. AM Si Cs[ ] Cs[ ] Cs[ ] () If he lniudinal eleial field is used ansi he sinal, hen he nea-field ase ( < ), elains f he speal pnens ae (ef Eq. ): whee ()
The speed a whih he wave up (envelpe) ppaaes an be deeined by squain he esulan dulaed sinal (Eq. ) and nin he ase shif ( ) f he dulain pnen ( ) f he esulan sinal. Pefin his puain n he lniudinal eleial field and usin inei ideniies yields: AM SiDe Cs[ ] ] Cs[ ] Cs[ = +O[ ] () The esulaan ase shif f he dulain pnen is:. Subsiuin he ase elains (ef Eq. ) yields: d The speed a whih he dulain envelpe (infain) ppaaes an hen be alulaed usin (Eq. ). Pefin his puain n he abve esul yields he sae answe as when alulaed usin he up speed (ef Eq. ): d Ne ha his del an als be used shw ha he nea-field eney densiy eneaed by an elei diple ppaaes a he up speed, sine he eney densiy (w) f an eleanei field is nwn be equal he su f he squaes f eah field pnen (p.7): w E E B (8) (6) (7) 6 Ppsed expeiens easue supeluinal nea-field ase and up speed f he lniudinal elei field The pevius aheaial auens have indiaed ha he up speed f a ppaain neafield lniudinal elei field is uh fase han he speed f lih f ppaain disanes less han ne wavelenh ( < ). The appxiae f f he up speed, f he nea-field lniudinal elei field (ef Eq. ) is: (9) The fllwin expeien is ppsed easue he nea-field up speed f a ppaain nea-field lniudinal elei field. I is suesed ha an apliude dulaed sinal be injeed in ne end f a paallel plae apai and deeed n he he side f he apai by an apliude dedula. The ase diffeene beween he esulan dedulaed sinal
and he iinal dulain sinal shuld hen be easued. The ap disane f he apai plaes shuld hen be ineased and he ase diffeene shuld hen be easued aain. The up speed an hen be deeined by enein he value f he dulain fequeny (), he easued hane in ase (), and he hane in apai plae ap disane (d ) in he fllwin elain: d () Beause he ase speed f he lniudinal elei field is uh fase han he speed f lih, he expeed ase hane ay n be easy easue. In he neafield, a disanes less han ne enh wavelenh (elaive he aie fequeny), he up speed f he lniudinal elei field is appxiaely (ef Eq. ): () d d d () Ne ha usin a hih dulain fequeny and an even hihe aie fequeny an inease he bseved ase hane. If a MHz dulain fequeny and a MHz aie fequeny wee used eneae he apliude dulain sinal hen a hane in apai plae disane wuld eneae a x - de ase hane. Ne ha hese fequenies espnd a 6 fa-field dulain eleial wavelenh and a.6 aie wavelenh. This ase hane wuld be vey diffiul easue bu i ay be pssible usin a hih ase sensiiviy l-in ehnique develped by he auh. 7 Supeluinal wave ppaain in he ysial syses 7. Eleanei fields eneaed by a anei diple. The eleanei fields eneaed by an sillain uen lp has been shwn by seveal auhs have a siila f he elei diple (p6) (pp. 6-6, 6). The nly diffeene beween he sluins is ha elei and anei fields ae evesed: Resulan eleial and anei fields f an sillain anei diple H Cs( ) i e i H Sin( ) i i e () E Sin( ) i i e ()
Cnsequenly, all he analysis pefed n he elei diple in he pevius paes an als be applied his syse. Speifially, he nea-field ase and up speeds f his syse ae als nluded be uh fase han he speed f lih. 7. Gaviainal fields eneaed by an sillain ass Maheaial analysis f an he aviainal fields eneaed by a vibain ass eveals ha f wea aviainal fields, aln he axis f vibain, he sillain aviainal diple is deled wih he a paial diffeenial equain siila ha f he sillain elei diple. One sinifian diffeene beween he w syses is ha in de nseve enu, a vibain ass us be apanied wih anhe sillain ass sillain wih he sae fequeny bu wih ppsie ase. The effe f he send vibain ass is anel he ppaain aviainal fields in he fa field. Hweve, nea he sue he send vibain ass des n nibue sinifianly he esulan ppaain aviainal fields, and an be neleed in he dellin 8. Usin he Einsein elain: 8G G T () Aln he axis f vibain, f sall asses and lw veliies, he Einsein equain edues : G T (6) The nly nn-vanishin e in he eney enu ens de is: d T dsin( ) O in whih: (7) Slvin he paial diffeenial equain yields 7 : K O l l Sin Cs (8) in whih: K = - G, d l, l d Sin (9) The aviainal field () an hen be alulaed usin he elain G d d : hh.. O O Sin ()
Sine d O, slvin his elain f he ase speed f he lniudinal aviainal field yields: d O () Theefe i is nluded ha nea he sue, aln he axis f vibain, he lniudinal aviainal ase speed is uh fase han he speed f lih. The up veliy has been shwn be (ef Eq. 8). Insein he elain f he ase speed (ef Eq. ) yields:. Insein he de esiae f he ase speed (Eq. ) yields: d O () F he abve esuls (Eq., ) i is nluded ha nea he sue bh he ase speed and he up speed f he lniudinal aviainal field, aln he axis f vibain f he ass, ae uh fase han he speed f lih. 8 Cnlusin This pape has pvided aheaial evidene ha eleanei nea-field waves and wave ups, eneaed by an sillain elei diple, ppaae uh fase han he speed f lih as hey ae eneaed nea he sue, and edue he speed f lih a abu ne wavelenh f he sue. The speed a whih wave ups ppaae (up speed) has been shwn be he speed a whih bh he wave eney densiy and dulaed wave infain ppaae. Beause f he siilaiy f he venin paial diffeenial equains, w he ysial syses (anei sillain diple, and aviainal adiain sillain ass) have been shwn have siila nea-field supeluinal esuls.
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