Analytical Formulas for Dipole Radiation Buried inside Planar Stratified Media

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1 6 t Ittol Cofc o AOSPAC SCINCS & AVIATION TCNOLOGY ASAT - 6 My 6-5 -Ml: st@tc..g Mlty Tccl Collg Koby lobb Co gypt Tl : Fx: + 69 Pp: ASAT-6--CM Alytcl Fols fo Dpol to B s Pl Sttf M S Tos Bsy * Os M. Abo-S St Sob Abstct: I ts pp lytcl fols v b v fo t lctogtc fls MFs t fo vtcl gtc pol VMD b pl sttf. T pl-ly coctg ol opt: swt go. xpssos fo t t vcto wc s to t t lctc gtc fl copots of t pol t t pl-ly c to t tgls obt by Sofl pvosly. A spl tcq wt t of t coplx g toy s s to obt gly cct lytcl xpssos fo Sofl tgls SI tt s ts clcltos wc l w vy tos coplct to vlt. Clos-fo xpssos fo t f fl t t gos to vtcl gtc pol b sco go s-wt clclt t pyscl gs scss. T slts clly vlt plott. I to t slts obt t t gos cop wt c ot wt tos to lsw pov tt ts to s vy ffctv spl. T fols copttos c b ppl to t cocto low fqcs go. T cv slts wll lso b sfl fo ot ssg of t oc sfc spclly w t tstt s clos to t sfc. Kywos: F-fl to t s lctogtc popgto sttf b vtcl gtc pol VMD Sofl Itgl SI. I. Itocto T lctogtc wv t fo vtcl gtc pol VMD b o sbg pl sttf s of tst fo ot ssg cocto pposs cowv systs ppl optcs. T pobl of pol to ov ws oglly folt by Aol Sofl s clssc wo pbls 99 []. Sc t ts pobl s cv sgfct ot of ttto s of pps v b pbls o t sbjct. T coplt stoy of t pobl s byo t scop of ts wo bt sffcs to sy tt pt vtos by Wyl [] Noto [] ots v pst o cct psttos fo t fl of t pol v cof tt t s o t sg of Sofl's pp []. I 94 Noto [4] s vlop s fol fo t fl copots to stg w cl slts c ly b obt. : * Cospog Ato: Pof. D. S T. Bsy Mttcs Dptt Fclty of Scc A Ss Uvsty Co gypt; stbsy@yoo.co Fclty of Scc Kf l-s Uvsty gypt; bos@sc.fs..g Cl g Isttt of gg & Tcology S.st@yoo.co

2 Pp: ASAT-6--CM I t pog wo by Wt [5 9]t Sofl tgl SI fo t lctogtc fl of lctc gtc pols tg t sttf ly go w vlt by sg syptotc toscoto tgto bc cts.i 966Bños [] pst coplt pspctv of t stocl vlopt of t ttcl lyss of t pobl. T lctogtc fls of vtcl ootl lctc pols o o t boy btw two fft v b wll ow ts of clos-fo xpssos fo y ys. W ts clsscl pobl s solv t Sofl tgl wll occ t s o wy of vog t vlto of t Sofl tgl fo t cl o clos fo soltos. D to t gly osclltoy slowly cy t of t Sofl tgl t ws ffclt to pply ct cl tgto tcq to vlt ts tgls. Tfo so wll-ow ppocs sc s stpst slpot to bc ct to sttoy ps pot to v b ploy to obt so ppoxt soltos. owv ts tos volvg lgty lgb svl tsfotos vy tos coplct. Moov t popts of t lctogtc fl t by pol gt o coplct w t boy cls ly of t tl wt tt popts. I t ly 9 s Cw Kog [] copt t lctogtc fl of ootl pol D o two-ly t. T ws ss to b low-loss sc tt t g-soc fls w pott. Itgl psttos of g soc fls w vlt wt fo syptotc ppoxtos. I 9 Bst [] v xpssos fo t fl copots of ootl lctc pol D t psc of coctg lf-spc s fols w vl fo t qs-sttc to t f-fl gs. I ss of wos by Kg [ 5] t coplt fols v b obt fo t lctogtc fls xct by ootl lctc pol D vtcl lctc pol VD pl two- t- ly. I 9 Cw [6] ws t fst wo v spl cct fol spl tcq fo Sofl tgl. Ts fol ws pst sly b stps. I Log t l. [7] vlop t spl tcq [6] v t f-fl ppoxto of t Sofl tgl ply pg o t sttoy ps-pot. Fto t slowly vyg pt t Sofl tgl s o sglts t t sttoy ps pot lg to s Cw's fols ot sl pobls to clclt t f-fl. T sbsqt vlopts o t lctogtc fl of pol ly go v b c ot by y vstgtos spclly clg Log t l. tcq [7]. ctly y pps w pbls by Bsy t l. [-] tt tt lytclly t lctogtc fl of vtcl gtc pol t psc of t-ly og go tls bcs of ts y sfl pplctos spclly costp t. I ts sc o to tp t scls w sbsttt t og go s pvosly Bsy t l. [-] wt pl. Tfo to obt g g of cl ffccy g cccy fst slts w s ts pp Log t l. [7] tcq wt t of t coplx g toy [] to v clos-fo xpssos fo t f fl t t pl ly to vtcl gtc pol b sco ly s wt. T pyscl g of ts xpssos s scss t cl copttos lso c ot. I to t slts pst gpclly cop wt tos to lsw. II. Gotc Dscpto Bsc Folto T lvt goty tt s llstt Fg. sows t ol s t clclto w t copss t lys s go. T soc s vtcl gtc pol VMD loct t s t l ly t pt ootlly. Pysclly ts

3 Pp: ASAT-6--CM psts sll loop t of S cyg ccltg ct I. s t stc btw t soc obsvg pot P wc s loct t s t pt. W sppos tt t s s of ft tcss. If w scb t goty of t syst by sg t cylcl pol coots t fo t wol spc. Moov t t go lys ft pw oww log t xs spctvly. All gos lso ft sw oogos. A ftl sspto ts lyss s tt t splct ct t s s glgbl cop to t cocto ct []. g I A II Swt V M D P III Go X Fg. : Gotc cofgto of VMD loct t s wt pl tfcs. It s wll ow tt t lctogtc fl gt by vtcl gtc pol sll ootl loop t c b v fo t gtc t vcto wc s oly o copot t -cto fo t t vto ; cosqtly t copots of t lctc gtc fls gv by []:. T tgt copots of t lctc o gtc fls cotos t t boy so t t vcto ts vtv by cotos []. T t -copot of t t vcto sol stsfy t followg boy cotos: t t = t = b Not tt t Sofl to cotos [] t fst t sol b stsf fo.

4 Pp: ASAT-6--CM 4 III T Itgl pstto of t Fls T M Accog to Sofl [] t -copot of t t vcto c go t t gl obsvto pot P c b obt s follows: A t C B t b D t c w = ts l pt s postv s t Bssl fcto of o o. A B C D t costt fctos wc w f t by cobg t boy cotos lg to: A 4 B 4b C 4c D 4 w j j j j j j s t Fsl flcto coffct. T o-vsg copots of t lctc gtc fls t t gos c b ctly obt ft f ot of pltv lgb by sbstttg fo s follows: I t go t : A 5 A 5b A. 5c

5 Pp: ASAT-6--CM 5 I t swt go t : C B o o 6 C B o 6b C B o 6c I t go go t : D 7 D 7b D. 7c It sol b ot tt t followg ottos toc: x x x x x 9 w o t sco l fcto of o o o spctvly. IV. Alytcl Solto fo t F-Fl A I ts scto w wll scss t pocss of solto fo t f-fl t go. I ts cs t obsvg pot P s t stc ov t s sfc. W focs o ttto o 5 to clclt t lctc fl. T gtc fls c b lso clclt sl wy. It s cl tt t tgl 5 psts t pw wv fo t s sfc to t obsvg pot. Sbstttg by t vl of t costt fcto A fo 4 5 t lctc fl c b wtt s follows: IS. Tg to ccot t followg ltos: x x

6 Pp: ASAT-6--CM 6 t t oto c b xpss s ft ss: IS. Accog to t coplx g toy [] w t coto s stsf w v: w. Usg t fols poply gg w v: 4 IS 4 w 5 5b 5c 4. 5 To f t vl of w s t tcq [7]. Tfo t tgl 5 c b wtt s:. 6 w t fst fcto 6 s ply vyg wl t sco o s slowly vyg. Moov ccog to os [] t locto of t sttoy ps pot s gv by: 7 wc s t followg solto:. W w c s t followg ppoxto fo l fcto:

7 Pp: ASAT-6--CM 4. 9 Aft sg 9 so stgt fow pltos q. 6 c b xpss s: By t s wy w c obt: 4 w t ppoxtos s bcs t coto Sbstttg by t vls of c b obt s follows: s stsf [] fo - to 4 t fl xpsso of IS N 4 w N T fl pot P s t stc t s sfc s llstt Fg.. I o to vlt t vl of t fl t s-wt go w st solv t tgl qtos 6. It s cl tt t fst t of t tgl s t ct wv t py fl fo t soc to t obsvg pot. Ts t s pott f t tstt cv st vy closly. owv f t two pots spt f fo c ot ts t wll vs bcs t popgto pt s t s t ttto s vy lg. T sco t s coc wt wc psts t oww wv fo t sfc of t 7 4. O of t cotbtos of ts pp s qto 4 w t copot of t lctc fl tt s to b clclt pvosly by Log t l. [7] wt ppoxto fo ow w tt ts vl wtot ts ppoxto. T pyscl g of q. 4 c b xpl tt t fst t cts ss of wvs wc popgt pw fo t soc o tps btw t s sfc t botto t tvl log t pt v t t fl pot w / s t flcto coffct t t s botto. T flcto coffct t t s sfc s g s lost ty. T sco t cts ot ss of wvs wc popgt oww fo t soc fst t flct pw t t s botto tvl log t pt v t t fl pot. V. Alytcl Solto fo t F-Fl S-Wt

8 Pp: ASAT-6--CM s to t obsvg pot. T lst t wc s t t t s coc wt tt psts t pw wv fo t s botto to t obsvg pot f t cv s clos to t botto ts t ss to b sfl. Sttg wt t tgl qto 6 ft sbstttg by t vls of t costt fctos B C fo 6b 6c w c wt t lctc fl s-wt ft f ot of pltv lgb s: o S o I 5 w 6 6b 6c Sttg wt t tgl q. 6 by sg t s to [7] w c wt s:. 7 W t fst fcto 7 s t ply vyg pt wl t sco o s t slowly vyg pt. T t locto of t sttoy ps pot s gv by: T solto of qto s: 9 Aft solvg t ply slowly pts w v t solto of s follows: I t s w obt t solto of s: 4 N b 4 c

9 Pp: ASAT-6--CM 9 w. Fo 5 t s sy to gt t solto fol fo t lctc fl s-wt s follows: N T pyscl g of c b xpl tt t sco t psts ss of wvs wc popgt pw fo t soc o tps btw t s sfc t botto t tvl log t pt v t t fl pot swt. T t t psts ot ss of wvs wc popgt oww fo t soc fst t flct pw t t s botto tvl log t pt v t t fl pot. VI. Alytcl Solto fo t F-Fl T Go I ts scto w wll ly t tgl qtos 7 to f t solto of t f fl t go go t-ly. I ts cs t fl pot P s t stc t s sfc. W bg wt t tgl qto 7 ft sbstttg t vl of t costt fcto D fo 4 t lctc fl c b xpss s: IS. Usg t fst fol of t t lctc fl c b wtt s: 4 Usg t sl pocs fo fg t fl slt of t lctc fl ft so lgty bt stgtfow pltos c lso b gv by: 4

10 Pp: ASAT-6--CM It s cl tt t fst t of 4 psts ss of wvs wc popgt pw fo t soc o tps btw t s sfc t botto t tvl log t pt v t t fl pot go. T sco t psts ot ss of wvs wc popgt pw fo t soc fst t flct oww t t s sfc tvl log t pt v t t fl pot. VII. Ncl slts W pst so gps sowg t lctc fls wc s copt t gos fo fft vls of tcss of t s. T soc s VMD loct s t gt bov t s botto. I ts Fgs t vtcl scl s ol by t fcto 5 4. To tt t cl clclto w sct tt s tc fo t cotos sow Fgs -. Fgs 5 sow clly tt t cs t s tcss l to cs t lctc fl. Moov Fgs 6 9 sow tt cs t gt of t soc css t lctc. Fgs 4 7 sow tt t lctc fl css w t vtcl stc of t fl pot s cs tl t cs to ct vl t cs. Fto Fgs - sow tt t lctc fl s stog s-wt go w t soc t fl pot t s go. Ts slts coc wt pvos slts obt by sg ot tos. owv ts to s c s lvs ts slts o qcly ts sv t t xst pvos tos ppocs. Tfo ts to wc s ot cotbto of ts pp s vy pott s t svs t sc's t fg t cct slts of t pobl vy sot t.

11 Pp: ASAT-6--CM

12 Pp: ASAT-6--CM

13 Pp: ASAT-6--CM

14 Pp: ASAT-6--CM VIII. Dscsso Coclso I ts pp w s o sc wo ts vsbl lvbl t coplt fols tt v b v fo t lctogtc fls gt by vtcl gtc pol b s t t-ly wt pl tfcs. T obsvto pot s loct swt go spctvly. Moov t stct pyscl gs of t v fls c of t t gos scss sptly. Ncl clcltos of t fl copots pfo llstt. Ts pp lso cotbts to t lyss of t MFs wc s bs o ts to s t ps o t s of so tgl tts t tfcto of t sttoy ps pot of t tg to vlt SI ply. Ts ws llstt by sowg t fo soltos of t f-fls w fst xp t wt t lp of t coplx g toy [] clos fo xpssos of t f-fl t t gos w obt sply. It s cssy to ot tt t slowly vyg pt SI s o sglts t t sttoy ps pot. T slts of ts wo g to gt xtt wt t covtol sttoy ps to o sl -pot to bt t s qc t c b sly s to clclt t f fls fo sl pobls. T slts obt sfl y gg pplctos sc s wt cocto cocto gopyscs xplog. It c lso b s vlopg fst spl tptto tools fo sttg t t stct ot ot ssg pplctos. IX. fcs [] Sofl Üb Asbtg Wll tlos Tlgp A. Pys vol. pp ; Vol. pp []. Wyl Asbtg ltogtsc Wll b b Lt A. Pys Vol. 6 pp [] K. A. Noto Popgto of o wvs ov pl t Nt Vol. 5 pp [4] K. A. Noto T clclto of go-wv fl tsty ov ftly coctg spcl t Poc. Ist. of o gs. Vol. 9 No. pp Dc

15 5 Pp: ASAT-6--CM [5].. Wt to fo vtcl lctc pol ov sttf go I Ts. Ats Popgt. Vol. No. pp [6].. Wt to fo vtcl lctc pol ov sttf go I Ts. Ats Popgt. Vol. No. 4 pp [7].. Wt to fo vtcl lctc pol ov cv sttf go ol of s. Nt. B. Sts Vol. 56 pp [].. Wt xctto of sfc wvs o coctg lctc cl cogt sfcs ol of s. Nt. B. Sts Vol. 59 No. 6 pp [9].. Wt lctogtc Wvs Sttf M Oxfo gl: Pgo Pss Nw Yo [] A. Bños Dpol to t Psc of Coctg lf - spc Pgo Pss Oxfo 966. [] W. C. Cw. A. Kog lctogtc fl of pol two-ly t Gopyscs Vol. 46 No. pp. 9-5 M. 9. [] P.. Bst T g toy of lctogtc fl of ootl lctc pol t psc of coctg lf-spc o Sc. Vol. 7 No. 5 pp. 95- Sp.-Oct. 9. []. W. P. Kg Nw fols fo t lctogtc fl of vtcl lctc pol lctc o coctg lf-spc ts ootl tfc. Appl. Pys. Vol. 5 pp Dc. 9; t Vol. 56 pp. 66 Dc. 94. [4]. W. P. Kg T lctogtc fl of ootl lctc pol t psc of t-ly go. Appl. Pys. Vol. 69 No. pp [5]. W. P. Kg T lctogtc fl of ootl lctc pol t psc of t-ly go: Spplt. Appl. Pys. Vol. 74 No. pp [6] W. C. Cw A qc wy to ppoxt Sofl-Wly-Typ tgl I Ts. Ats Popgt. Vol. 6 No. pp Nov. 9. [7] Y. Log. g B. bol F-go lctogtc to wt Vtcl Mgtc Dpol S I Ts. Ats Popgt. Vol. 49 No. 6 pp [] S. T. Bsy O. M. Abo-S. S. Sob lctogtc to Fls T-Ly M wt og Itfcs I Ts. Gosc. ot Ssg Vol. 5 No. pp [9] S. T. Bsy O. M. Abo-S. S. Sob Scttg fo Ly-Stcts wt og Bos lctogtcs Vol. No. 7 pp [] S. T. Bsy O. M. Abo-S. S. Sob Wv Popgto A fo Vtcl Mgtc Dpol Loct T og-ly Stcts ol of lctogtc Wvs Applctos MWA Vol. 7 No. 7 pp [].. Wt lctogtc fls of socs lossy.. Coll F.. c s.. At Toy. Pt II. Cp. 4 pp McGw-ll. Nw Yo [] D. S. os T Toy of lctogts Pgo Pss Loo 964. [] A. Sofl "Ptl Dfftl qtos Pyscs" Loo U.K.: Acc Pss 964. [4] S. T. Bsy s wt t Dptt of Mttcs Fclty of Scc A Ss Uvsty Co 566 gypt -l: stbsy@yoo.co. [5] O. M. Abo-S s wt t Dptt of Mttcs Fclty of Scc Kf l- S Uvsty Kf l-s 56 gypt -l: bos@yoo.co. [6]. S. Sob s wt t Dptt of Mttcs Cl Isttt of gg & Tcology S gypt -l:.st@yoo.co.