PLASMONIC WAVEGUIDES FEATURES CORRELATED WITH SURFACE PLASMON RESONANCE PERFORMED WITH A LOW REFRACTIVE INDEX PRISM

U.P.B. Sci. Bull., Sris A, Vol. 75, Iss. 4, 013 ISSN 13-707 PLASMONIC WAVEGUIDES FEATURES CORRELATED WITH SURFACE PLASMON RESONANCE PERFORMED WITH A LOW REFRACTIVE INDEX PRISM Gorgiana C. VASILE 1, Aurlian A. POPESCU, Mihai STAFE 1, S.A. KOZIUKHIN 3, Dan SAVASTRU, Simona DONŢU, L. BASCHIR, V. SAVA 4, B. CHIRICUŢĂ 4, Mona MIHĂILESCU 1, Constantin NEGUŢU 1, Nicula N. PUŞCAŞ 1 Th surfac plasmon rsonanc calculations wr prformd for Krtschmann configuration. Two of th layrs, gold and chalcognid films (GaLaS), hav finit thicknss. Th rfractiv indx of coupling prism matrial is supposd to b lowr than th rfractiv indx of amorphous film which composs a plasmonic wavguid. Th amorphous film thicknss for which plasmonic rsonancs occur was stablishd. Th calculations wr prformd for TE- and TM- mods (both for visibl and IR domain) concrning th propagation constants, th attnuation cofficint and th lctromagntic fild distribution. Th rsults provid th conditions for dsign an optical mmory dvic basd on light-light intraction in plasmonic configuration. Kywords: plasmonics, photonics, chalcognid amorphous matrials, GaLaS 1. Introduction Th plasmons concpt was introducd sinc th bginning of th XXth cntury as an intraction of lctromagntic wavs with th plasma lctrons from solid matrials (mainly mtals) which contain fr lctrons. Sommrfld (1899) and Znnck (1907) hav xplaind th confind propagation phnomnon of lctromagntic wavs on th mtals surfac. Th phnomnon analysis was ddicatd to wirlss tlgraph opration, opration basd on low frquncy radio wavs. In th optical band th confinmnt of lctromagntic wavs at dilctricmtal intrfac was xplaind by E. Krtschmann and H. Rathr [1] in 1968, who hav proposd and xprimntally ralizd a nw xcitation way of th surfac 1 Univrsity POLITEHNICA of Bucharst, Physics Dpartmnt, Splaiul Indpndnti, 313, 06004, Bucharst, Romania Corrsponding authors: A.A. Popscu: apopscu@ino.ro, M. Staf: staf@physics.pub.ro, National Institut R&D of Optolctronics, INOE 000, 409 Atomistilor str., PO BOX MG. 5, 7715 Magurl, Ilfov, Romania 3 Kurnakov Institut of Gnral and Inorganic Chmistry RAS (Russia, Moscow), Lninsky prospct 31, Moskow, 119991, Russia 4 S.C. Apl Lasr SRL, Vintila Mihailscu Str, No. 15, Bl. 60, Sc. 1, Et. 1, Ap. 1, Sct. 6, Bucharst, Romania

31 Gorgiana C. Vasil t al. plasmons using vanscnt wavs which form in total rflction conditions or prism mthod. Confinmnt of light propagation is obtaind on th mtal-air intrfac of th mtal film dposd on th prism, whn th light is dirctd on th prism at a rsonanc angl. Th thortical modl usd by ths authors study a singl intrfac of two smi-infinit mdia and is basd on solving of th Maxwll quations in transvrs-magntic (TM) and transvrs-lctric (TE) cass and applications of continuity conditions []. This analysis lads to th disprsion quation ω ( k) for th surfac plasmon-polariton. Th xamination of th disprsion quation obtaind in this cas can provid svral conditions ncssary to raliz th confinmnt of light: a) th incidnt light on th prism must b polarizd in th incidnc plan calld transvrs magntic (TM); b) dilctric constant (ral part) of th film must b ngativ; c) th dilctric constant of th mtal must xcd th dilctric constant of th matrial which is in contact with th film. This matrial is usually th ambint (air) or dissolvd substancs in watr. Du to th tight confinmnt of light on th mtal film surfac, this configuration has found many applications in dvloping of som optolctronics snsors basd on plasmon rsonanc [3-5]. As was dmonstratd arlir, th surfac plasmon rsonanc (SPR) is vry snsitiv, up to 10-4 - 10-5 changs of th rfractiv indx. Th xcllnt rsults wr obtaind in plasmon rsonanc applications as biosnsors [6, 7]. A nw dirction in th surfac plasmon rsonanc applications consists in th dposition on th mtal surfac of a dilctric film whos proprtis can b changd by applying an lctric fild, illumination s.a. In th chalcognid glasss (ChG) changs of th optical constants aris undr th action of light with th photon nrgy xcding th band gap. Stationary and dynamic photoinducd absorption may b distinguishd [8]. Th modifications of th optical absorption corrlat with th modifications of th rfractiv indx by Kramrs Kronig rlation. Howvr, th chang of th rfractiv indx in chalcognid films is small, on th ordr of 10-3, and a snsitiv xprimntal mthod has to b usd. Th surfac plasmonic rsonanc is an opportunity to dvlop nw dvics. In a rcnt invstigation th authors [9] proposd and hav xprimntally dmonstratd th SPR configuration light modulation by us of thin GaLaS chalcognid light snsitiv amorphous which is dpositd on top of silvr film with th thicknss of 30-50 nm. In ordr to satisfy plasmonic rsonanc conditions thy us a TiO high rfractiv indx prism. Using th transfr matrix formalism, in [10] hav bn lucidatd svral rsonanc angls in SPR structur which contain a mtallic and amorphous chalcognid films with high rfractiv indx ( n =. 47 ), typical for GaLaS or As S 3 compounds. In this papr th cas of GaP prism with rfractiv indx highr ( n = 3. 3) was xamind. Th rsults ar xplaind by coupling of vanscnt wav with plasmonic wavguid mods.

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 313 Th nd to us th high rfractiv indx matrials for th prism manufactur limits th typ of usabl matrials. Th fulfilling of this condition is ncssary if on considr th total xcitation spctrum of th mods. Th possibility to us th matrials with lowr rfractiv indx of th prism, for xampl BK7 glass ( n = 1. 51), for xcitation mods of As S 3 plasmonic wavguid (matrial with high rfractiv indx) has bn stablishd in [11]. In this papr only th numrical simulations rsults for TE mods wr prsntd. Th TM mods hav not bn xamind du to th complx charactr of th propagation constant and th complx numbrs opration. In th prsnt papr w invstigatd both TE and TM mods by numrical simulations. W dtrmind th structur paramtrs for which th coupling of fr lctromagntic wav to th confind mods of th planar plasmonic wavguid can b ralizd.. Th plasmonic wavguid structur with GaLaS chalcognid film Th typical plasmonic planar wavguid consists of a structur of long and larg layrs in th plan yz. In ach rgion th rfractiv indx may b considrd as constant. Th constitunt rgions ar as follows: - a substrat mad from oxid glass (BK7) which may b considrd smiinfinit; - a thick nough gold film which hav a complx rfractiv indx; - a dilctric film mad of GaLaS which is th wavguid; - a covr rgion considrd smi-infinit (air). Th modl of th structur that carry rsonanc light coupling in plasmonic wavguid is shown in Fig. 1. Hr, n m dnots th ral part of th rfractiv indx of th mtallic layr. Th rfractiv indxs of th layrs at two wavlngths ar givn in Tabl 1. On can s that th optical wavguid layr has th largst indx n f > n m > n a. According to th matrix calculations, th thicknss of th gold film could b around 50 nm in ordr to achiv 100% rsonant coupling of th incidnt radiation to th wavguid. W may tak into account also that du to skin-ffct th pntration dpth of light in gold is about 0 nm. It mans that in visibl and nar IR spctral domain th mtallic film can b considrd to b thick nough so as th substrat matrial didn t significantly altr th wavguid mod spctrum. In th sam tim th wavguid fild may hav a wak coupling with vanscnt wav on th prism bas. Manwhil th wavguid fild may hav a wak coupling with th vanscnt wav on th prism.

314 Gorgiana C. Vasil t al. Fig.1. Plasmonic wavguid structur and th rflctiv indx profil. Th ral valu of th complx mtal rfractiv indx is prsntd in Fig. 1. Th covr (air) and substrat (BK7) rgions ar much thickr than λ and w considr ths rgions as smi-infinit. Such an approximation considrably simplifis th form of th disprsion quation and also th finding of solutions in th complx plan. Finally, w will writ th lctromagntic fild rlations for th 3-layr wavguid systm as th filds ar wakly linkd with th prism mdium. Th paramtrs valus that wr usd in calculations ar shown in th tabl blow. Tabl 1. Th layrs rfractiv indx at two working wavlngths No. Incidnt lasr wavlngth (nm) Air rfractiv indx GaLaS rfractiv indx [13] Gold rfractiv indx [1] *) 1. 633 1.48 0.197-3.09i. 1310 1.38 0.48-8.75i *) W us th convntion (n-iκ) for th complx rfractiv indx as th plan wav blow ar takn in th form of [ i( βz ωt )]. Th mtallic substrat lis in th rgion x<0 and th air covr lis in th rgion x>d. Th film thicknss d is varid btwn 0.1 and 1.6 μm which is comparabl to th lasr wavlngth. 3. Light propagation in plasmonic wavguids: numrical simulation for TEmods cas Th lctric filds of th propagating TE mods in th thr rgions can b writtn as follows: ik x ik x iβz Ey wavguid( x, z) = ( A + B _ ) (1a) E E k x iβz ( x, z) = C 3 k ( ) ( x d ) iβz x, z = D 1 y _ mtalfilm (1b) y _ covr (1c)

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 315 Hr, k = nk0 β is th transvrsal wav numbr within th optical wavguid along th x dirction. Th wav numbrs for th covr and th mtallic rgions ar givn by th rlations k1 = β n1 k0 and k3 = β n3 k0, rspctivly. λ0 = π / k 0 = c ω is th wavlngth of th incidnt radiation. Th propagation constant β along th z dirction is th sam in th thr layrs of th wavguid du to th continuity condition of th lctromagntic fild componnts at th wavguid boundaris. Th disprsion quation for th TE mods can b obtaind from th continuity conditions of th lctric fild Ey and of th magntic fild H z (proportional to th drivativ x E y of lctric fild) at th two boundaris of th wavguid. As w ar intrstd to th rlativ fild distribution, w may considr on of th intgration constant qual to unity. By xcluding th othr constants, th following quation may b obtaind: idk ( ik + k1)( ik + k3) = () ( ik k1)( ik k3) Taking into account that : idk 1 tanh( idk ) = and tanh( idk idk ) = i tan( dk ), (3) 1+ th following rlation btwn transvrs wav numbrs k i ( i = 1,, 3) may b obtaind: k ( ) ( ) k1 + k tan k 3 d = (4) k k1k3 Equation (4) constituts th disprsion rlation k i = f ( ω). For th convninc of numrical solution, Eq. (4) may b invrtd and writtn in th form: k k k d = 1 + + mπ k 3 arctan arctan k (5) In Eqs. (3-5), m rprsnts th mod numbr and d is th film thicknss, which play th rol of paramtr. This quation rlatd to losslss wavguids was obtaind by Marcus [14, 15] for th first tim. All th wav numbrs war considrd as rals. In our cas of contrary, th transvrsal wav numbr as wll as th propagation constant β should b considrd as complx numbr and suitabl solution mthods may b applid.

316 Gorgiana C. Vasil t al. (c) (d) Fig.. () (f) E fild of diffrnt TE mods across th thr rgions of th wavguid for a film y thicknss of 0.5 μm (a, b) and 0.5 μm (c-f) at 633 nm wavlngth.

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 317 Fig. 3. (c) E y fild of diffrnt TE mods across th thr rgions of th wavguid for a film thicknss of 0.5 μm and 0.5 μm (b, c) at 1310 nm wavlngth Th numrical solutions of Eq. (5) and th lctromagntic filds in th thr rgions givn by Eq. (1) wr dtrmind by using Matlab. Th rsults corrsponding to λ 0 =633 nm ar prsntd in Figs., 4 and 6, whras th rsults for λ 0 =1310 nm ar prsntd in Figs. 3, 5 and 7. Th lctric fild E y across th wavguid (i.. as a function of transvrs x dirction) for th TE propagating mods at λ 0 =633 nm is givn in Figs. (a, b) for d=0.5 μm, and in Fig. (c-f) for d=0.5 μm, rspctivly. For incidnt radiation at 1310 nm wavlngth, th lctric fild E y across th wavguid for th TE propagating mods is givn in Figs. 3 for d=0.5 μm, and in Fig. 3 (b, c) for d=0.5 μm. On can obsrv that th numbr of propagating mods is two tims smallr in cas of 1310 nm wavlngth as compard to 633 nm radiation.

318 Gorgiana C. Vasil t al. (c) (d) () (f) Fig. 4. Intnsity of th TE mods in xz sction of th wavguid for a film thicknss of 0.5 μm (a, b), and 0.5 μm (c-f) at 633 nm wavlngth. (c) Fig. 5. Intnsity of th TE mods in xz sction of th wavguid for a film thicknss of 0.5 μm and 0.5 μm (b, c) at 1310 nm wavlngth. Th light intnsity distribution in xz cross sction of th wavguid for th two propagating mods (TE 0 and TE 1 ) in th 0.5 μm wavguid is givn in Figs. 4(a, b). Th 0.5 μm thick wavguid nabls propagation of four TE mods, namly TE 0 -TE 3. Th light intnsity distribution in xz cross sction of th wavguid for ths mods is givn in Figs. 4(c-f). Th light intnsity distribution in xz cross sction of th wavguid for th propagating mod (TE 0 ) in th 0.5 μm wavguid is givn in Fig. 5. Th 0.5 μm thick wavguid nabls propagation of two TE mods, namly TE 0, TE 1. Th light intnsity distribution in xz cross sction of th wavguid for ths mods is givn in Figs. 5(b, c).

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 319 Fig. 6. Th ral and imaginary parts of th ffctiv rfractiv indx as a function of wavguid thicknss, for th first four TE mods at 633 nm wavlngth. Fig. 7. Th ral and imaginary parts of th ffctiv rfractiv indx as a function of wavguid thicknss, for th first thr TE mods at 1310 nm wavlngth. Figs. 6 and 7 prsnt th ral and imaginary parts of th ffctiv rfractiv indx β ko of th wavguid as a function of wavguid thicknss for diffrnt TE mods at 633 nm and 1310 nm wavlngth radiation. Th disprsion curvs in Fig. 6 indicat that th 0.5 μm thick wavguid nabls propagation of two TE 0 and TE 1 mods at 633 nm wavlngth, whras Fig. 7 indicats that only on mod (i.. TE 0 ) can propagat at 1310 nm wavlngth. A closr analysis of ths figurs dmonstrats that th ffctiv indx of th TE mods incrass asymptotically from 1 to.48 at 633 nm (rspctivly.37 at 1310 nm wavlngth), i.. btwn th smallst rfractiv indx (th covr s) and th highst rfractiv indx (th film s), whn incrasing th wavguid thicknss. By analysing th Figs. 6 and 7 on can s that th attnuation of th propagating mods is strongr for th propagating mods at 1310 nm as compard to 633 nm. For xampl, for th thicknsss 0.5 and 0.5 μm, th imaginary ffctiv rfractiv indx (which also givs th attnuation cofficint along th z

30 Gorgiana C. Vasil t al. dirction) for th TE 0 mod is approximatly four tims smallr at 633 nm than 1310 nm wavlngth. This can also b sn in Figs. 4 and 5. (c) (d) () (f) Fig. 8. Th ral (a-c) and imaginary (d-f) parts of th ffctiv rfractiv indx as a function of incidnt wavlngth, for diffrnt propagating TE mods. Th influnc of th incidnt lasr wavlngth on th propagation of th TE mods in wavguids of diffrnt thicknsss is givn in Fig. 8. Th dpndnc of rfractiv indxs n = n f and n 3 = nm of th dilctric film and mtallic layr on wavlngth is takn from [1, 13]. Th figur indicats that for th full rang of wavlngths undr study hr (500-1400 nm) th ffctiv indx dcrass with wavlngth. Morovr, th figurs dmonstrat that th TE 0 mod can propagat in wavguids thickr than 0.5 μm (Fig. 8), TE 1 mod nds thicknsss largr than 0.5 μm (Fig 8), and TE mod nds wavguid thicknsss largr than 1 μm (Fig 8(c)). Figs. 8 (d-f) show that th attnuation of th TE mods along th z dirction bcoms strongr as th wavlngth incrass.

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 31 4. Light propagation in plasmonic wavguids: numrical simulation for TM-mods cas Th lctric filds of th propagating TM mods in th thr rgions can b writtn as follows: ik x ik x iβz H y wavguid( x, z) = ( A + B _ ) (6a) H H k x iβz ( x, z) = C 3 y _ mtalfilm (6b) k ( ) ( x d ) iβz x, z = D 1 y _ cov r (6c) Th disprsion quation for th TM mods can b obtaind from th continuity conditions of th magntic fild H y, and of lctric fild E z componnts at th two boundaris of th wavguid. Prforming th sam stps as abov, th disprsion rlation may b obtaind: n k n k k d = 1 + 3 arctan arctan + mπ n k n k (7) 1 3 In Eqs. (6,7), m rprsnts th mod numbr and d is th film thicknss. Th form of th quation is th sam as was obtaind in [14] whn th mdia wr considrd losslss, maning that th matrial paramtrs wr considrd ral. Basd on th plasmonic structur prsntd in Figs. 1 a), b) th numrical simulations wr prformd in Matlab and som rsults ar prsntd in Figs. 9, 10, 11 and 1. Th distribution of th magntic fild H y across th thr rgions of th wavguid (i.. th transvrs x dirction) for th TM propagating mods is givn in Figs. 9 (a,b) for d=50 nm and in Figs. 9 (c-f) for th 500 nm film thicknss, rspctivly. Th mtallic film lis in th rgion x<0 and th air covr lis in th rgion x>d.

3 Gorgiana C. Vasil t al. (c) (d) () (f) Fig. 9. TM mods for a film thicknss (d) of 50 nm (a,b) and 500 nm (c-f) Figs. 9 (a, c) indicat a sharp spik of th magntic fild of TM 0 mod at th film-mtal intrfac. In othr words, thr is a strong confinmnt of th mod in ~50 nm thicknss of film nar th film-mtal intrfac. In fact, TM 0 mods corrspond to surfac wav mods. Thrby, it is xpctd that th TM 0 mod propagation and attnuation to b practically indpndnt of film thicknss whn thicknss is largr than ~100 nm. This is dmonstratd in Fig. 10, which prsnts th ral and imaginary parts of th ffctiv rfractiv indx of th wavguid as a function of wavguid thicknss for th TM 0 mod. Fig. 10 also indicats that th 50 nm thick wavguid nabls propagation of two mods: TM 0 and TM 1 mods. Th light intnsity distribution in xz cross sction of th wavguid for ths mods is givn in Fig. 11 (a,b). Th 500 nm thick wavguid nabls propagation of four TM mods, namly TM 0 - TM 3. Th light intnsity distribution in xz cross sction of th wavguid for ths mods is givn in Fig 11 (c-f).

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 33 Fig. 10. Th ral and imaginary parts of th ffctiv rfractiv indx as a function of wavguid thicknss d, for th first four TM mods. A closr analysis of Fig 10 a) dmonstrats that th ffctiv indx of th TM 1 -TM 3 mods incrass from 1 to.48, i.. th rfractiv indics of covr and film. Th ffctiv rfractiv indx corrsponding to th TM 0 mods incrass rapidly with film thicknss to a valu largr than th.48. Fig. 10 indicats also that th attnuation of th TM 0 mod along th z dirction is ~ 1 ordr of magnitud highr than for th othr mods. This is consistnt with th rsults prsntd in Figs. 11 (a,c) which dmonstrat th TM 0 mod propagat on distancs of th ordr of 400 nm along th z axis, whras Fig. 11 (b,d-f) dmonstrat that th othr mods propagat ovr distancs of th ordr of 4 μm.

34 Gorgiana C. Vasil t al. (c) (d) () (f) Fig. 11. Intnsity in xz sction of th wavguid of th TM mods for a film thicknss of 50 nm (a,b) and 500 nm (c-f) 5. Conclusions Th invstigations of th guidd mods providd th numrical simulations of plasmonic wavguid structur. Th structur is composd by gold film, chalcognid GaLaS amorphous film of finit thicknss and smi-infinit covr, air for instanc. Th analysis showd th fild distributions, attnuation for diffrnt mod and ffctiv indx valu ncssary for coupling th light into plasmonic wavguid via prism with low rfractiv indx. For xampl, th film thicknss must b in a narrow rgion of 10-180 nm (Fig.10) in ordr to xcit TM 1 mod. Or, in a narrow band from 180 to 40 nm (Fig. 6) at th wavlngth 633 nm for coupling of light into TE 1 mod. A spac rsolution as low as 400 nm may b ralizd for th TM 0 mod. Th rsolution incrass at longr wavlngth in th IR rgion.

Plasmonic wavguids faturs corrlatd with surfac plasmon rsonanc prformd... 35 Acknowldgmnts. This work was supportd by a grant of th Romanian National Authority for Scintific Rsarch, CNDI UEFISCDI, projct numbr PN-II-PT-PCCA-011-5 / 01. R E F E R E N C E S [1] Krtschmann, E. and Rathr, H. (1968). Radiativ dcay of non-radiativ surfac plasmons xcitd by light. Z. Naturforschung, 3A: 135-136 [] Stfan A. Mair, Plasmonics: Fundamntals and applications, Springr, 007. [3] Srgiy Patskovsky, Andri V. Kabashin, Michl Munir, John H.T. Luong, Nar-infrard surfac plasmon rsonanc snsing on a silicon platform, Snsors and Actuators B 97, (004), 409-414 [4] Zdzislaw Salamon, H. Angus Maclod, Gordon Tollin, Surfac plasmon rsonanc spctroscopy as a tool for invstigating th biochmical and biophysical proprtis of mmbran protin systms. I: Thortical principls, Biochimica t Biophysica Acta 1331, (1997), 117-19 [5] Jirı Homola, Sinclair S. Y, Guntr Gauglitz, Surfac plasmon rsonanc snsors: rviw, Snsors and Actuators B, Vol. 54, (1999), 3-15 [6] Yun-Tzu Chang, Yuh-Chun Lai, Chung-Tin Li, Chng-Kuang Chn, Ta-Jn Yn, A multifunctional plasmonic biosnsor, OPTICS EXPRESS, Vol. 18, No. 9, (010), 9561-9569 [7] Rajan Jhaa, Anuj K. Sharmab, Dsign of a silicon-basd plasmonic biosnsor chip for human blood-group idntification, Snsors and Actuators B Vol. 145, (010), 00-04 [8] J. Ornstin, M. Kastnr, Photocurrnt Transint Spctroscopy: Masurmnt of th Dnsity of Localasd Stats in a-as S, Phys. Rv. Ltt. Vol. 46, (1981), 141-144 3 [9] Zsolt L. Sámson, Shih-Chiang Yn, Kvin F. MacDonald, Knton Knight, Shufng Li, Danil W. Hwak, Din-Ping Tsai, Nikolay I. Zhludv, Chalcognid glasss in activ plasmonics, Phys. Status Solidi RRL, Vol. 4, (010), 74-76, [10] A.A. Popscu, R. Savastru, D. Savastru, S. Miclos, Application of vitrous as-s-s chalcognids as activ layr in surfac plasmon rsonanc configuration, Digst journal of nanomatrials and biostructurs, Vol. 6, no 3, (011), 145-15 [11] Gorgiana C.Vasil, Roxana Savastru, A.A. Popscu, M. Staf,D. Savastru, Simona Dontu, L. Baschir, V. Sava, B. Chiricuta, M. Mihailscu, C. Ngutu and N.N. Puscas, Modlling Th d Plasmonic StructursWith Activ Chalcognid Glass Layr, Romanian Rports in Physics, Vol. 65, No. 3, (013), 101 1018 [1] Edward D. Palik, Handbook of Optical Constants of Solid, Acadmic Prss, Boston, 1985. [13] Marvin J. Wbr, Handbook of Optical Matrials, CRC PRESS, 003. [14] D. Marcus, Coupling Cofficints For Imprfct Asymmtric Slab Wavguids, Bll Syst. Tch. J., Vol. 5, (1973), 63-8 [15] D. Marcus, Thory of dilctric optical wavguids, Scond d., Acadmic Prss, Inc., 1991