Leaky modes in multilayer uniaxial optical waveguides

Leaky modes in mul ilaye uniaxial op ical wa eguides Lluis To ne , Fe nando Canal, and J. He nandez-Ma co The p opaga ion cha ac e is ics o he leaky modes in plana aniso opic wa eguides wi h a mul ilaye s uc u e ha e been in es iga ed by means o a compac igo ous o malism. The leakage losses and leaky ansi ion angle ha e been s udied o he undamen al and i s hyb id modes. An inhomogeneous wa e- guide and bu e ed s ep index ype s uc u e ha e been discussed. Pa icula a en ion has been de o ed o he a ia ion o he loss coe icien o he leaky modes as a unc ion o bu e hickness and bu e e ac i e in- dex. A no ably di e en beha io has been ob ained o a ious con igu a ions. Keywo ds: Leaky modes, wa eguide heo y, in eg a ed op ics. 1. In oduc ion A g ea numbe o bo h ac i e and passi e in eg a ed op ical de ices a e ab ica ed by using uniaxial aniso- opic ma e ials, such as LiNbO 3o LiTaO 3. Elec- oop ic modula o s, swi ches, mode con e e s, cou- ple s, and pola iza ion con olle s ha e been made on i anium-di used li hium nioba e.12The de ailed knowledge o he p ope ies o he aniso opic op ical wa eguiding s uc u es appea s highly use ul o im- p o e he op imiza ion le el o such de ices, and, also, i may sugges a new class o de ice based speci ically on wa eguide aniso opy. 3 Fo homogeneous h ee-laye s uc u es exac ana- ly ical solu ions a e a ailable in he li e a u e. This p oblem, in addi ion o he inhomogeneous case, has been analyzed wi h g ea de ail by se e al au ho s 2 who ha e poin ed ou he in e es ing p ope ies o such wa eguides. In pa icula , leakage losses we e heo- e ically desc ibed and expe imen ally measu ed.13"1 4 The bidimensional case has also been s udied by means o he coupled-mode heo y1 5and he e ec i e index me hod.'6Howe e , because o he ising pe - o mances equi ed by he inc easingly sophis ica ed op ical ci cui s, he e has been in ecen yea s g owing in e es in mul ilaye wa eguides. These s uc u es a e widely used since hey p o ide mo e choices o he designe in addi ion o i s pa icula p ope ies. Con- ce ning he leaky modes in such s uc u es, in e es in hem comes om hei connec ion wi h he mode- The au ho s a e wi h Poly echnic Uni e si y o Ca alonia, Depa - men o Signal Theo y & Communica ions, Apdo. 30 002, 08080 Ba celona, Spain. Recei ed 13 Oc obe 1988. 0003-6935/90/182805-10$02.00/0. © 1990 Op ical Socie y o Ame ica. dependen loss wa eguides and aniso opy based cu - o de ices. Ne e heless, al hough he heo y o modal dispe sion in mul ilaye iso opic wa eguides has been e y well desc ibed, i s aniso opic coun e - pa is no ye ully de eloped. In his pape we use he ans e -ma ix me hod o es ablish he wa eguiding condi ion o ligh p opaga- ion in plana uniaxial dielec ic wa eguides wi h a mul ilaye s uc u e wi h no es ic ions on he op ical axes o ien a ions. This me hod p o ides a e y com- pac o malism o analyze such s uc u es, and i has been ex ensi ely used in he iso opic case.1 7I is based on he well known 4 X 4 o malism, which has been de eloped by Be eman'8and Vassell'9and in a di e en way by Yeh.20 A new o mula ion was e- po ed ecen ly by Knoesen e al. 2' and o special cases by Walpi a. 22 The app oach can be summa ized as ollows. In he homogeneous subs a e and supe - s a e he o al ields a e a supe posi ion o he o di- na y and ex ao dina y wa es p opaga ing in a uniaxi- al unbounded medium. These solu ions a e con inued ac oss he in e media e dielec ic medium by means o a cha ac e is ic ma ix con aining he ield solu ions. Thus he equi ed bounda y condi ions on he op and bo om wa eguide in e aces a e exp essed by a de e - minan al condi ion which yields o he wa eguiding condi ion. The p ocedu e can be applied o he exac analy ical s udy o uniaxial mul ilaye s uc u es and he nume ical analysis o inhomogeneous uniaxial wa eguides. De ails o he analysis a e gi en in Sec. II. As an applica ion o he o malism, in Sec. III we in es iga - ed he leaky-mode p opaga ion in bo h an inhomoge- neous wa eguide and a s ep index s uc u e wi h a dielec ic bu e laye . The leakage losses o he wa e- guide ha e been s udied o he undamen al and i s hyb id modes o di e en op ical axis o ien a ions. In he i s case, we mainly de o ed ou a en ion o he 20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2805 e ec s in oduced by he inhomogenei y. In he sec- ond case, he s udy was ocused o he a enua ion coe icien o he leaky modes and he guided- o-leaky mode ansi ion angle as a unc ion o he bu e hick- ness and bu e e ac i e index. II. T ans e -Ma ix App oach An asymme ic con igu a ion o uniaxial c ys als o ming a h ee-laye s uc u e wi h a bi a ily o ien - ed op ical axes is assumed (Fig. 1). In he p incipal axis coo dina e sys em o he c ys als, he dielec ic enso akes he o m (EO ) (1) Supe s a e DT Subs a e [E], [Ei (Z) [E] S z Fig. 1. Schema ic wa eguiding mul ilaye s uc u e. P opaga ion is along x. eo and e, being he o dina y and ex ao dina y pe mi - i i ies, espec i ely. Fo a gi en o ien a ion o he op ical axis () de e mined by he pola and azimu hal angles (0,sp), he componen s o he dielec ic enso a e ob ained om Eq. (1) by means o he applica ion o he associa ed o a ion ans o ma ion (70,(p). Thus one ob ains exx = e0(sin 2(p + COS2p COs 2 O) + e cos2Sp sin20, = O(cos 2(p + sin2V cos20) + ee sin2 sin20, EZZ = e sin20 + E Cos20, (2) xy = (e -e) sin p cos sin20, exz = (e e,) cosp sinO cosO, eyz = (Ee -) sinip sinO cosD. In a wa eguide such as he one desc ibed abo e only hyb id modes can p opaga e excep o specially sym- me ic &-axis o ien a ions. I he p opaga ion di ec- ion is aken o be along he x-axis and assuming ime ha monic dependence, he ields a any poin ha e he o m exp[i(x -w )], being he p opaga ion cons an . Ou side he guiding laye he ields mus be e anescen so ha E,(z) = E, exp(Ycz), z < 0, (3) E8(z) = E, exp[y9(D-z)], z > D, (4) whe e he subsc ip s c and s s and o supe s a e and subs a e egions, espec i ely. By subs i u ion o Eqs. (3) and (4) in o he wa e equa ion, V2E + pW 2[e]E = V(V E(5) and aking in o accoun Eq. (2), an homogeneous equa- ion sys em is ob ained, he de e minan o which mus anish. This condi ion leads o wo possible solu ions o he decay cons an s y,, ha co espond o he o dina y and ex ao dina y wa es Y 0=:I: /32-_ w2e 0(6) 'Ye = -2( 2Z) -AC0 2eoeeEzz()i -, . (7) He e he uppe sign in he pa en heses holds o he subs a e and he lowe o he supe s a e. Acco d- ing o he ans e -ma ix me hod, a any pai o ans- e sal planes Z 1,Z2 he angen ial componen s o he ields can be ela ed by using a ma ix T, which may be de ined as = T[E . (8) The exis ence o his ans e ma ix is gua an eed by he linea i y o he Maxwell equa ions. In ac , he ma ix T con ains he ield solu ions in he egion be ween he planes Z1,Z2, so ha in Eq. (8) i con inues he solu ion om z o Z2ac oss he in e media e e- gion. Le us ake z1 =0 and Z2= D. Thus he equi ed bounda y condi ions o he angen ial ield compo- nen s a e necessa ily e i ied i he ield solu ion is (8) join wi h he known solu ions (3)-(4) in he subs a e and co e . This condi ion can be w i en in a ma ix o m using he ollowing s anda d p ocedu e. The exis ence o solu ions (6) and (7) equi es ha all he componen s o he ields be exp essed in e ms o wo o hem. Then we ha e chosen as independen a i- ables he 9 and x componen s o he elec ic ield associa ed wi h he o dina y and ex ao dina y wa es, espec i ely. In his way, he elec ic ield in he su- pe s a e (z < 0) can be w i en as E,(z) = 1Ey exp(-yz) + Ay E exp(-yez), (9) 20ZO -Aze- and in a mo e compac no a ion Ax, I exP(z)} E,(z) [; A:] {Exp PYez) (10) Fo he magne ic ield one has H,(Z) [yoAxo aZO ye ze Exe exp(-yz j (11) LU 43 i 3Aye Jxe exp(yez) J The coe icien s A appea ing in hese exp essions a e ob ained om Eq. (5). Again in z < 0 one has 2806 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990 - ...l A2 1 0=_ -Y -iyy' {E 297 +,3( -E2) + 2ie,y,68 A 20 E2(MwE 2 2 -iy i) _ ey 2(,Y2 + gco 2Ej 2 x=(Co -E )-Y2 + 2(e +2 AZO(e ),2+ /20 _ e ) + 2i z ~ ' (12) -AN34 - AeN24 -eA>eN 2j - [N34 --AN14 -'yAN1 y] + N24 -<N14 -4yeAN1 2] + Y[N23-N1,3 + AyN 12l = 0, (24) (13) whe e 2 py 22) + E(zA2L'z -i3)( Aye }'X(,Y - 2 + .2 )(h2 z _ 02) _ (W2,Ey)2 (14) (2 _ j2 + PW2E 2 )( 22) + A 2)2 Az= -+ ,' -Y (15) (y2 _ 2 + pU2E )(pue2-2) -(2e , 2 On he o he hand, acco ding o Eq. (4), he co e- sponding exp essions o he subs a e w i e in he same way as Eqs. (10)-(15) by making he ans o ma- ion -y ---y o bo h he o dina y and ex ao dina y wa es. Thus one has i * 1 E; 0exp[%(D-z)] Ez) = 1 Aye J (16) [Az*, A,*, JEexp [*(D- z)I H,(z) = + iA 0) 0(,y + i#A;e) ExO exp[((D -z)] IE*, exp[y(D - z)] (17) Th oughou he supe sc ip * indica es a subs a e pa ame e . Subs i u ing Eqs. (10), (11) and (16), (17) in o Eq. (8) an homogeneous equa ion sys em is ob- ained which can be w i en as Axo I p1o 7,le O Te '2o D2e R e | 1 A, >'3o V3e Eyo _ -, -,ye '4o V4J E,, (18) Nj jol!e V joV (25) Equa ion (24) has been ob ained o a bi a y wa e- guide pa ame e s wi h no es ic ion on he op ical axis o ien a ion o he c ys als. Then i is no iceable ha o he o ien a ions which allow sepa able TE-TM ield solu ions Eq. (24) is no ably simpli ied. This si ua ion happens when one (o bo h) subs a e and supe s a e is (o a e) iso opic (i.e., e = ee) and also when he op ical axis lies in he same plane as he p opaga ion di ec ion and is con ained in he plane pe pendicula o ha o he wa eguide (in ou no a ion s = 00). As well, when 5s = 0 = 900, he ield solu ions allow a TE-TM decomposi ion. In he second case, when he c-axis lies in he (P = 00 plane in bo h he subs a e and supe s a e, he exp essions o he E and H ields a e iden ical o Eqs. (10), (11), (16), and (17), bu now Axo = 0, Aye = 0, Az, = 0, and $LW 22 -i 'ze= )2 _- W z (26) The wa eguiding condi ion is ob ained as o he gen- e al case. Thus i can be w i en in a o mallyiden ical way o he iso opic s uc u e 9in e ms o he new a iables: EoEe 1 EZZ 8 _ #X2 ¢ 2 =Y+i zE (27) (28) (29) whe e O 2 Ayo + iAL, (19) e; + i/Aze, (20) j {il -1,3, (21) 11W=,2,4, wi h 12j2 TjhAx, + Tj3 7- ±j2 (, -'y7j4), (22) Q2- Tjl + T12AYe -(Tj2Te -eAyeTj4)- (23) As is explici in hese exp essions, he j unc ions mus be e alua ed in he supe s a e. The condi ion o ha ing a non i ial solu ion in Eq. (18) is ha he de e minan o he coe icien s anishes. This condi- ion leads o a es ic ion o he possible alues o he p opaga ion cons an ,, which mus now o m a dis- c e e se , and, he e o e, i is he wa eguiding condi- ion. A e s aigh o wa d manipula ion i can be exp essed as and simila exp essions o e*, an and 2e (Appendix A). I is well known ha 5 6"9'21 i he op ical axis o he uniaxial media o ming he guiding laye lies also in he (,o = 0° plane, he wa eguide suppo s pu e TE and TM modes. As usual, in his case Eq. (24) spli s in o wo (one TE and one TM) e y simple exp essions. Conce ning he po = = 90° case, i can be conside ed as well by aking he co esponding exp essions o he iso opic case bu wi h e = e0in he TM e ms and e = e in he TE ones. Likewise in a g ea numbe o p ac i- cal applica ions he supe s a e is iso opic and he subs a e is a uniaxial c ys al. In his case one also has Ax = 0, Aye = 0, Az, = 0, and Aze = -iI3/yc, whe e _y = V#2 kgn 2, (30) nc and ko being he e ac i e index o he iso opic medium and he ee-space wa enumbe , espec i ely. Then he wa eguiding condi ion is gi en by Eq. (24), bu now Eqs. (22) and (23) simpli y o Lj. = Tj3 + Ž Tj4,(31) 20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2807 Qje = T+ n Tj2, (32) The ans e ma ix o an homogeneous uniaxial dielec ic slab can be analy ically calcula ed by using he 4 X 4 o malism. In he gene al case, i con ains a supe posi ion o he wo o dina y and wo ex ao di- na y wa es p opaga ing in a bi e ingen medium. In he case o mul ilaye s ep index wa eguides, he ans e ma ix o he s uc u e is ob ained by he p oduc o all he ma ices associa ed wi h each sub- laye , in acco dance wi h he o de ing c i e ion p e- sc ibed in Eq. (8). When di used wa eguides a e conside ed, in which he inhomogenei y akes place in addi ion o he aniso opy, he p oblem becomes mo e complica ed. In such a case a closed- o m solu ion o Maxwell equa ions is no gene ally a ailable, and he associa ed ans e ma ix mus be e alua ed wi h he help o app oxima e me hods o nume ical echniques. The la e can be made h ough he di ec in eg a ion o Maxwell equa ions in he di usion egion by means o such s anda d nume ical p ocedu es as he Runge- Ku a me hod" o he Gea p edic o echnique, 16 which a e a ailable in mos ma hema ical lib a ies. Also, he well known mul ilaye s ai case echnique can be use ul in analyzing especially complex s uc- u es con aining homogeneous addi ional laye s. In his case he inhomogeneous egion is conside ed as a ini e se o hin homogeneous ilms, o each he ans- e ma ix is analy ically known. Finally, since we ha e imposed no es ic ions on he exis ence o complex alues o he p opaga ion con- s an 3, he abo e de i ed exp essions a e applicable o bo h guided and leaky modes. Thus i is wo h no ic- ing ha he squa e oo s appea ing in Eqs. (6) and (7) o he ans e se p opaga ion cons an a he sub- s a e and supe s a e (oye) mus be e alua ed wi h he igh sign consis en wi h he beha io o mode ields a om he wa eguide. Special a en ion is equi ed when conside ing leaky modes due o hei imp ope na u e. 02 23 Ill. Discussion As an applica ion o he o malism de eloped in he p eceding sec ion, we ha e analyzed he e he p opaga- ion cha ac e is ics o he leaky modes o a ious mul- ilaye wa eguides ab ica ed in dielec ic uniaxial ma- e ials. Such modes come om he us a ion on o al in e nal e lec ion a he in e aces be ween he guiding laye and su ounding dielec ic media and a e ob ained as complex solu ions o he eigen alue equa- ion. Conce ning he aniso opic case, wi h a sui able o ien a ion o he c ys al op ical axes, one o he pola - iza ions (o dina y and ex ao dina y) su e s leakage losses, whe eas he o he emains guided in he ilm. These kinds o mode. a e leaky guided modes in con- as o he leaky unguided modes which occu in he iso opic case.24 We ha e conside ed he e he case in which he op i- cal axes o he c ys als lie in he wa eguide plane (0 = 900), making an angle ep wi h he posi i e i-axis wi h he same alue in all uniaxial media. The eigen alue equa ion associa ed wi h his case is ob ained om Eq. (24) by aking in o accoun he sui able cha ac e is ic ma ix o he s uc u e. The explici exp ession o his cha ac e is ic ma ix is gi en in Appendix B. A wa eguide such as he one desc ibed abo e can suppo only hyb id modes wi h he six ield componen s. When he op ical axis makes a. small angle wi h he wa eguide axis (o -00) he ield componen s associa - ed wi h he o dina y and ex ao dina y wa es a e weakly coupled and he modes co espond o he TE- TM pola iza ions. When he angle inc eases, he cou- pling be ween he abo e componen s g ows also and se e al modes become leaky. The na u e o hese leaky modes is desc ibed in Re s. 7-10. He e we dis- cuss some open ques ions om which new esul s come. A. Inhomogeneous Wa eguide Fi s , we ha e analyzed an asymme ic inhomogene- ous wa eguide wi h a Gaussian p o ile in bo h he o dina y and ex ao dina y e ac i e indices, and in which he co e is in ai . The a ious wa eguide pa- ame e s a e Inq = 2.2946 ne = 2.21081 G1:. nos = 2.2866 nes = 2.2028 I n= D=21im He e D is he cha ac e is ic dep h o he Gaussian p o ile, and he subsc ip s and o ilm pa ame e s. Also we assumed X = 633 nm. When s = 0, G, suppo s he TEO and TMO modes. The mode which is he TEO mode a so = 0 emains guided o all alues o so and becomes he TMo mode a p = 90° ia a p edomi- nan ly o dina y hyb id mode. On he o he hand, he mode ha a s = 0° is TMo becomes leaky beyond n 110, and i con e s in o he TEO a so = 900 by means o a p edominan ly ex ao dina y hyb id mode. These modes a e e e ed o as he [TEo,TMo] [g] and [TMoTEo] [1], espec i ely. He e [g] and [] indica e a guided and leaky mode. We deal now wi h he beha - io o he loss coe icien o he [TMoTEo] [] mode. In Fig. 2 we ha e plo ed he loss coe icien o he abo e men ioned leaky mode as a unc ion o he angle so. To pe o m he calcula ions use has been made o a nume ical zoom oo - inding algo i hm o sol e di ec - ly he eigen alue equa ion (24) o complex oo s. Al- hough his p ocedu e equi es a conside able numbe o i e a ions in he complex -plane, i p o ides e y accu a e esul s. The g aded index p o ile has been in oduced in he o malism h ough he mul ilaye s ai case echnique. Fi s , he accu a e nume ical e- sul s ob ained o a ew alues o he angle (o a e gi en in Table I oge he wi h he esul s epo ed by Ko- shiba e al.'2 om he ini e elemen echnique. As can be seen, he ag eemen be ween bo h se s o alues is qui e good, he di e ences being unsigni ican in p ac ice. On he o he hand, ou main aim in his case is o examine he e ec s o inhomogenei y on he loss coe - icien . The eupon, we included also in Fig. 2 he 2808 APPLIED OPTICS / Vol. 29, No. 18 / 20 June 1990 T Sol 1P (deg ees) Fig. 2. Loss coe icien o he [TMo,TEo][lJ mode suppo ed by Gi as a unc ion o he op ical axis o ien a ion. ol is he guided- o- leaky mode ansi ion angle. Table I. Accu a e Nume ical Values Ob ained o he Wa egulde wi h a Gaussian P o ile (G1) and Compa ison wi h he Fini e Elemen Resul s Repo ed In Re . 12 'p T ans e Ma ix Fini e Elemen Re(Q/ko) Loss (dB/cm) Re(//ko) Loss (dB/cm) 120 2.28576 170.1 2.28574 181.6 300 2.26757 54.0 2.26756 53.8 600 2.22573 8.9 2.22572 8.9 esul s co esponding o a wa eguide iden ical o GI bu wi h a s ep index p o ile. This plo dese es some commen s. Fi s , in he case o he homogeneous p o- ile he loss coe icien shows a seconda y maximum which is no obse ed o he Gaussian p o ile (see also Fig. 3). Also, he s ep index wa eguide shows a p o- nounced loss peak which does no appea in he inho- mogeneous case. These ea u es ag ee wi h he p e i- sions o Bu ns e a.8in he sense ha he sha p s uc u e showed by he loss coe icien in a s ep index wa eguide is due o in e e en ial phenomena which o igina e in he ab up discon inui ies a he ilm- co e and ilm-subs a e in e aces. The same con- clusions ollow om Figs. 4 and 5, whe e he loss coe i- cien o he [TMOTEo] [1] mode has been plo ed as a unc ion o he co e e ac i e index o wo di e en alues o p. In all cases, o emphasize he e ec s due o he Gaussian p o ile he cha s ha e been e e ed o he alue o he loss coe icien o NC = 1. Fi s , i can 40- o gaussian 20 _ s ep-index 30 60 90 P° (deg ees) Fig. 3. De ail o Fig. 2 showing he seconda y maximum o he loss coe icien as a unc ion o so o he homogeneous wa eguide and he mono onous dec ease which occu s o he Gaussian p o ile. be seen in bo h igu es ha he loss coe icien depends on n, in a s onge way o he Gaussian p o ile han o he s ep index one. Also, in he case o he homoge- neous wa eguide, he beha io o he loss coe icien on nc is comple ely di e en o qp = 300 han o p = 40°. This di e ence, which does no occu o he Gaussian p o ile, comes again om in e e en ial phenomena. Finally, i is in e es ing o no e ha he p esence o he g aded index p o ile modi ies he alue o he c i ical angle a which he [TMoTEO] [1) mode becomes leaky. As men ioned abo e (ij 11° o he Gaussian p o ile, whe eas spo 140 o he homogeneous wa e- guide. B. Mul ilaye S ep Index S uc u es We deal now wi h mul ilaye uniaxial s uc u es. The in e es in such wa eguides comes om hei anal- ogy wi h hei iso opic coun e pa s, whose no iceable p ope ies ha e been poin ed ou in se e al wo ks. Ou main mo i a ion on his subjec lays in he ac ha he p esence o addi ional laye s modi ies no only he eal pa o he e ec i e index [ieG(/ko)] bu also he loss coe icien o he leaky modes and he guided- o-leaky mode ansi ion angle. In highly asymme ic s uc u es he p opaga ion cha ac e is ics o he leaky modes a e only sligh ly modi ied by he p esence o addi ional laye s a he op o he wa eguide. This is because in hese wa eguides he leaky modes mainly accoun o adia ion o he subs a e, so ha he co e ma e ial only a ec s he leaky modes when i s e ac i e index amoun s o a alue close o ha o he subs a e. The si ua ion changes comple ely when s uc u es wi h a high deg ee o symme y a e conside ed. Now he leaky modes 20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2809 207 2.18.12. =400 -o ~ ~ - n 15o gaussian h os o iie c=1 0 15 _10- / 0 s e-ine 1.7 1.9 2.1 2.3 nc Fig. 4. Loss coe icien o he leaky mode in Fig. 2 as a unc ion o he co e e ac i e index o ex = 40' To emphasize he e ec s due o he Gaussian p o ile he cu es ha e been e e ed o he alue o he loss coe icien o n oe 1. =300 10 'I M25- 0 '~gaussian 0 ~~~~4 ~~~~s ep-index 1.7 1.9 2.1 2.3 n, Fig. 5. Same as in Fig. 4 bu o (p = 300. co espond o adia ion owa d bo h subs a e and supe s a e, so ha hey a e s ongly dependen on co e pa ame e s. 12 Fo example, his is he beha io shown in Figs. 4 and 5, which we ha e discussed p e i- ously. The o malism de eloped in Sec. II allows us o analyze in an exac and simple way he mul ilaye uniaxial wa eguides. As al eady men ioned, he cha - X 2.187 [TMITEo[g| Q) 2.1[5 TEoTMo] g] 'TM:/ 1 - TEo TMO 2.183- I I I I I 0 30 60 90 P (deg ees) Fig. 6. E ec i e indices o he hyb id modes suppo ed by G2as a unc ion o *,. A s = 0,900 he modes a e pu e TE and TM. The [TMo,TEo] [g] mode is a p edominan ly ex ao dina y guided hyb id mode, whe eas he [TEO,TMo] [1] mode is a p edominan ly o dina y leaky hyb id mode. Solid line: pu e guided mode. Dashed line: leaky mode. ac e is ic ma ix o he whole s uc u e is ob ained by he o de ed p oduc o he ma ices o each laye . As an example, we ha e analyzed a s ep index symme ic wa eguide in which co e , subs a e, and ilm a e as- sumed o be uniaxial media. In addi ion, he e is an iso opic bu e laye o hickness Db and e ac i e index nb placed be ween he co e and ilm. Opposi e he o me case, we conside now a LiTaO3based wa e- guide wi h he ollowing pa ame e s: n( = 2.1856 ne/ = 2.190 G2: n = 2.1834 nes = 2.1878 D = 2 jim In he limi ing cases so = 0 and so = 900 his wa eguide suppo s he TEo and TMo modes. Fo any o he alue o he angle '1, G2suppo s wo hyb id modes, which will be deno ed as [TMoTEo][g] and [TEo,TMoJ[1] acco ding o he no a ion in oduced abo e. The guided- o-leaky mode ansi ion angle o he leaky mode amoun s o p 28°. The olls o he TE and TM modes ha e been in e changed in ela ion o he o me case (GI) because now we a e dealing wi h a posi i e bi e ingen ma e ial (ne > no). Figu e 6 shows he p opaga ion cha ac e is ics o he abo e modes as a unc ion o . The in luence o he bu e laye on he [TEo,TMoI[1] mode has been analyzed in Figs. 7-11. In Fig. 7 we plo ed he loss coe icien o his mode as a unc ion o he bu e hickness, and he a ia ion o he loss coe icien wi h he bu e e ac i e index is shown in Fig. 8. Two di e en o ien a ions o he c ys al op ical axis ha e been conside ed: = 300 and 2810 APPLIED OPTICS / Vol. 29, No. 18 20 June 1990 20 2.189 0 ep = 30° 0 2.10 2.15 2.20 Fib Fig. 7. Loss coe icien as a unc ion o he bu e e ac i e index. Mode [TEo,TMoJ [1]. Bu e hickness: Db = 0.5 ,pm. 0 u0 T4 Log[Db/A] Fig. 8. Loss coe icien as a unc ion o he decimal loga i hm o he A-scaled bu e hickness. Mode [TEoTMoJ[1j. Bu e e ac i e index: nb = 2. s = 400. The cha s in hese igu es show ha he e ec s o he addi ional laye on he loss coe icien o he leaky mode depends s ongly on he bu e hick- ness and is p ac ically insensi i e o he bu e e ac- i e index. In his case, he ob ained beha io o he loss coe icien is simila o bo h c ys al o ien a ions. Howe e , his esul canno be gene alized. Fo exam- ple, in Figs. 9 and 10 he p e ious dependences ha e 15- o=300 n ~ ~ ~~~n W~~~ 10 2.14 2.16 2.18 2.20 Fig. 9. Same as inFig. 7 o he [TE,,TMZJ mode suppo ed by he mul imode e sion (D 6 pm) o 02. 10 5- p=30 0 5~~~~~~~~~ ~~/ Log[Db/A] Fig. 10. Same as in Fig. 8 o he [TE,TM [ mode suppo ed by he mul imode e sion (D = 6pum) o G2- been plo ed o he [TETM 1J [1J mode suppo ed by a mul imode e sion (D = 6 im) o G2. In ac , his mul imode wa eguide also suppo s he [TM1,TE][U] mode, bu i is a e y low loss leaky mode.7As can be seen in Figs. 9 and 10, now he e ec s due o he bu e laye a e comple ely di e en o he wo conside ed c ys al o ien a ions. Finally, i is o be emphasized ha he p esence o 20 June 1990 / Vol. 29, No. 18 / APPLIED OPTICS 2811 u' C;' W . Log[Db/A] Fig. 11. Guided- o-leaky mode ansi ion angle o he [TEo,TMoI (1] mode suppo ed by G2as a unc ion o he decimal loga i hm o he ;-scaled bu e hickness. Dashed line: ansi ion angle in he absence o he bu e laye . he dielec ic low index laye also modi ies he alue o he guided- o-leaky mode ansi ion angle. Fo exam- ple, in Fig. 11 we plo ed he a ia ion o he leaky ansi ion angle wi h he bu e hickness o a ious alues o he bu e e ac i e index o he [TEoTM 0] [1] mode suppo ed by G2. Simila beha - io has been ob ained o o he leaky modes. As in Figs. 7 and 9, he leaky ansi ion angle becomes insen- si i e o he bu e hickness when Db amoun s o a alue o he o de o X. The modi ica ion o he guid- ed- o-leaky mode ansi ion angle w ih he wa eguide pa ame e s should be pa icula ly in e es ing o he aniso opy based cu o de ices in which his angle plays a undamen al ole. We e u n o his ques ion in a o hcoming pape .25 IV. Concluding Rema ks The wa eguiding condi ion o ligh p opaga ion in plana dielec ic uniaxial wa eguides wi h a mul ilaye s uc u e has been ob ained wi h no es ic ions on he op ical axes o ien a ion. The p ocedu e is based on he ex ension o he ans e -ma ix o malism o in- clude an aniso opic subs a e and supe s a e. As an applica ion o he o malism we ha e in es iga ed he p opaga ion cha ac e is ics o leaky modes in an inho- mogeneous wa eguide and a mul ilaye s ep index s uc u e in which he c ys al op ical axes lie in he wa eguide plane. The explici exp ession o he e- qui ed cha ac e is ic ma ix is also gi en. Bo h he loss coe icien and he guided- o-leaky mode ansi- ion angle ha e been analyzed as a unc ion o he inhomogenei y and bu e laye pa ame e s. We ha e shown ha he p esence o addi ional low index laye s o ming mul ilaye s uc u es wi h a high deg ee o symme y s ongly a ec s he p opaga ion cha ac e - is ics o he leaky modes suppo ed by such wa e- guides. These e ec s depend also on he op ical axis o ien a ion due o in e e en ial phenomena which come om he p esence o ab up discon inui ies on he in e aces be ween he a ious dielec ic media. The au ho s a e g a e ul o Ji i C y oky o he Czechoslo ak Academy o Sciences, P ague, o alu- able sugges ions. This wo k was pa ially suppo ed by a g an o he CYCyT o he Spanish Go e nmen (PB-87-0798-C0302). Appendix A The wa eguiding condi ion o he case when he - axis lies in he plane so = 0° in bo h he subs a e and supe s a e is {21 + i T22 + T11 -2 *T12 X {T 4+jŽAwT, + i2:wT 33 -2 T 34 1 { ( * X { 2 T1 -e E*ye T, {T 41+ i--T4 2 +Z--g T32 = -. 'ye A'. &' J(Al) When he wa eguide suppo s pu e TE and TM modes, he i s b acke in he abo e exp ession co e- sponds o he TM pola iza ion and he second o he TE one. In such a case, also, he o he wo b acke s anish iden ically. Appendix B The ans e ma ix o he bu e ed uniaxial wa e- guide in Sec. III is gi en by T = UaUb, whe e Ua and Ub a e he ma ices associa ed wi h he aniso opic ilm and bu e laye , espec i ely. Bo h ma ices can be analy ically calcula ed by means o he 4 X 4 o mal- ism. The de i a ion ollows a gene al p ocedu e de- eloped by Vassell, 19 who has also epo ed he exp es- sion o he ans e ma ix o he simple case o a uniaxial dielec ic ilm whose op ical axis lies in he P -00 plane. The ans e ma ix o he gene al case can be easily calcula ed also by means o he same p ocedu e.26 In ou case, 0 = 900, i can be w i en as Um = (X2a2 + A o) jm (Bi) U being an auxilia y ma ix whose elemen s a e gi en by 2812 APPLIED OPTICS I Vol. 29, No. 18 / 20 June 1990 l = p 0cos(wDX0) + X2aC cos(wDXe), 12 = e in(wDX,) + ai (e sin(wDXe)i U 3=- 01X[CEosmDX) +'~ U13 = a1A[cos(wDX)- cos(wDXe)], U14 = -ialki[sin(DX) -, sin(wDXe) Tg21 = i d [Aeo sin(wDX 0) + a2a Xe sin(wDXe)], X" 23 = oal sin(oDX,) - x sin(wDXe)I U24 = -pcaj[coswDX) -cos(wDXe)], U33= a1 cos(coDX 0) + Ac6cos(wDX,), U34 = -i E [ 1o~ea2 sinQwDX 0) + peo sin(wDXe)], X e U43 = -i [Xoa sin(wDX0) U22 = ll 044 = p 33, U 31 = U24 32 = l4, U41 = U23 42 = _U13, whe e Xe = Pee a e£ -exx cxy Xe .1 He e a = ,1/co and D is he hickness o he aniso opic ilm. The ans e ma ix o he iso opic bu e laye comes di ec ly om he abo e exp essions by making he subs i u ion E, = = enb, e being he ee space pe mi i i y and nb he bu e e ac i e index. One a i es a he well known exp ession Ub= (UTM 0U ) (B7) whe e [ cos(koDbb) i(i/a) sin(kODb) (B8) =i(azn) sin(keDb b) cos(koDb b) J He e X =_1i7; and Db s ands o he bu e hickness. Also, use has been made o he de ini ion ib" -N (B9) N= - /k0being he e ec i e index and aTE = -0b,aTM = Re e ences 1. L. Thylen, "In eg a ed Op ics in LiNbO3: Recen De elop- men s in De ices o Telecommunica ions," IEEE/OSA J. Ligh wa e Technol. LT-6, 847-861 (1988). 2. E. Voges and A. Neye , "In eg a ed-Op ic De ices on LiNbO3 o Op ical Communica ion," IEEE/OSA J. Ligh wa e Technol. LT-5, 1229-1238 (1987). 3. T. K. Gaylo d and A. Knoesen, "Passi e In eg a ed Op ical Aniso opy-Based De ices," J. Mod. Op . 35, 925-946 (1988). 4. D. P. Gia Russo and J. H. Ha is, "Wa e P opaga ion in Aniso- opic Thin-Film Op ical Wa eguides," J. Op . Soc. Am. 63, 138-145 (1973). 5. W. K. Bu ns and J. Wa ne , "Mode Dispe sion in Uniaxial Op ical Wa eguides," J. Op . Soc. Am. 64, 441-446 (1974). 6. D. 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