Applicability of the.Lorentzian peak method to analyze leaky and lossy optical waveguides

RAPID COMMUNICATIONS Applicabili y o he.Lo en zian peak me hod o analyze leaky and lossy op ical wa eguides LIuis To ne , Josep Fe e , and Fe nando Canal Poly echnic Uni e si y o Ca alonia, Depa men o Sig- nal Theo y & Communica ions, Apdo. 30002.08080 Ba ce- lona, Spain. Recei ed 29 Janua y 1991. Sponso ed by Elsa M. Ga mi e, Uni e si y o Sou he n Cali o nia. 0003-6935/91/182418-03$05.00/0. © 1991 Op ical Socie y o Ame ica. Se e al conclusions conce ning he applicabili y o he Lo en zian peak me hod o compu e he complex p opaga- ion cons an o leaky and lossy wa eguide modes a e es ab- lished. Usually he calcula ion o he loss coe icien o leaky and lossy modes appea ing in common in eg a ed op ical ech- nology cons i u es a e y cumbe some and long p ocess. This di icul y is a esul o such modes being complex guid- ed ones, so ha hei p opaga ion cons an exhibi s bo h eal and imagina y pa s. To ind he e ec i e index o hese modes (N = N + jNi, complica ed anscenden al equa- ions ha e o be sol ed in mos cases. This poin equi es a high numbe o i e a ions in he complex plane, and as a consequence a g ea olume o calcula ions a e gene a ed. Recen ly, an app oxima e p ocedu e, which is e e ed o as he Lo en zian peak me hod (LPM), has been epo ed1 o analyze his p oblem. This me hod is based on he ac ha o he cases o p ac ical in e es he imagina y pa o he e ec i e index amoun s o e y small alues. Typically N ~ 1 and Ni ~ -4,-5. Un o una ely, he e is no clea c i e ion o es ablish he applicabili y o he me hod o a pa icula s uc u e. The main aim o his wo k is o add ess his ques ion by means o a sligh ly di e en app oach han he o iginal p ocedu e. The eigen alue equa ion o a gene al wa eguiding sys em can be w i en as D(N) = 0, D(N) being a anscenden al complex unc ion. Acco dingly, he e ec i e indices o he modes suppo ed by he wa eguiding s uc u e a e ob ained as he eal and complex ze os o ha unc ion. So, i Nm = N m + jNim is he e ec i e index o a gi en mode, one has D(Nm) = 0, and in he neighbo hood o Nm he unc ion D(N) can be app oxima ed as D(N) ≃ A(N - Nm), A being a complex cons an . Then, in oducing he unc ion (N ) ≡ |D(N ,Ni = 0)|2, which is only de ined along he eal N-axis, one ob ains (N ) ≃ |A|2[(N , - N m)2 + N2im. In a i s app oach (LPM), he unc ion 1/ (N ) can be plo ed o ob ain Lo en zian beha io . Using he ac ha Nim « 1, i can be assumed ha app oxima e alues o N m and Nim can be ob ained by cu e- i ing he esul ing plo .1 Howe e , usually he alues ound o N m and Nim using his p ocedu e depend on he sample ange used in he cu e- i ing p ocess. This di icul y can be a oided by aking in o accoun ha (N ) can be w i en also as 2418 APPLIED OPTICS / Vol. 30, No. 18 / 20 June 1991 Fig. 1. Loss coe icien as a unc ion o he op ical axis o ien a ion o he s ep index wa eguide ( = 2 μm). We ha e no included he s ongly inaccu a e LPM esul s a θ ≃ θc. Now, he N m and (N m) a e known, his equa ion de ines a unipa ame ic eal unc ion, and Nim can be s aigh o wa d- ly calcula ed by a simple linea eg ession p ocedu e. Also, i allows us o ob ain app oxima e alues o N m and (N m), since (N ) exhibi s a ela i e minimum a N = N m. This is a single and objec i e alue, i.e., i does no depend on he used sample ange, and hus ep oducible esul s a e ob- ained. The amoun o calcula ion equi ed is simila o wha is needed in he usual oo - inding schemes o he pu e guided modes (Nim = 0), since he i e a ion p ocedu e can be pe o med by a as con e ging minimizing algo i hm. To check he applicabili y o he abo e p ocedu e we ha e conside ed he o -axis p opaga ion in a ypical X,Y-cu LiNbO3-based wa eguide, in which he c ys al op ical axis lies in he wa eguide plane making an angle θ wi h he wa eguide axis. The co esponding eigen alue equa ion has been de i ed om he s anda d ans e -ma ix o malism.2 Fi s , we conside a s ep index wa eguide o hickness in which he co e is in ai . The wa eguide pa ame e s a e no = 2.2946, ne = 2.2108, nos = 2.2866, nes = 2.2028, = 2 μm, and λ - 633 nm. He e nos,nes a e, espec i ely, he o dina y and he ex ao dina y e ac i e index o he subs a e, whe eas no ,ne co espond o he ilm. When θ = 0°, his wa eguide suppo s he TE0 and TM0 modes. The TE0 mode a θ = 0° emains guided o all alues o θ; meanwhile he mode which is he TM0 a θ = 0° becomes leaky beyond θc ≃ 14°. In Fig. 1 we plo ed he loss coe icien o his mode as a unc ion o θ. The con inuous line co esponds o he exac solu ion ob ained by di ec ly sol ing he eigen alue equa ion by means o a nume ical zoom oo - inding algo- i hm. The dashed line ollows om he modi ied LPM. Table I. Inhomogeneous Wa eguide The ag eemen be ween bo h se s o alues is excellen , e en in he egion o highes losses. Also, he e o in he calcula ion o N m amoun s o a negligible alue in mos cases, so ha i is only no iceable in he egion whe e Nim has g ea alues. This is because, gene ally speaking, o he poin s belonging o he eal N- axis, he linea app oxima ion o D(N) becomes less accu- a e when he ac ual oo mo es away om his eal axis. The e o is educed o a negligible alue by making a second calcula ion by means o he new unc ion ƒ1(N ) ≡ |D(N ,N0im) N0im being he alue ob ained a he i s s ep. Also, i is wo h no icing ha he LPM does no wo k in he egion θ ~ θc, since his is a cu o poin and he ac ual oo s s em om a sha p s uc u e o D(N) in he complex plane. On he o he hand, e y good ag eemen has also been ob- se ed o a mul imode e sion ( = 3 μm) o he abo e example. This ac has o be emphasized, since in his case he loss coe icien shows a complica ed beha io as a unc- ion o θ wi h a ious maxima and minima.3 Now, we a e going o examine an inhomogeneous wa e- guide wi h a Gaussian p o ile in bo h no and ne , which suppo s also a pu e guided mode and a leaky guided mode. The wa eguide pa ame e s a e iden ical as in he o me single-mode example. Again, he accu acy ob ained in his case using he modi ied LPM is e y good when Nim o he leaky mode has small alues. Ne e heless, he esul s in he egions wi h mode a ely high losses s ongly disag ee wi h he exac alues.2 Table I shows some calcula ed al- ues. This impo an disag eemen occu s in a ange o ~25° beyond θc ≃ 11° and comes om he pa icula o m o he unc ion D(N). In Fig. 2 he shape o he unc ion (N ) in he neighbo hood o N m is shown o θ = 80°. The plo has wo minima. The i s one, which occu s o he smalle alue o N , is a local minimum and does no co espond o any oo o he equa ion D(N) = 0. The emaining mini- mum, which occu s a a highe alue o N (on he igh o he s eep maximum), is he one associa ed wi h he ac ual ze o o he unc ion D(N) and leads o he esul s gi en in Table I. The shape o unc ion (N ) shown in Fig. 2 is he same, wha e e he alue o θ. Ne e heless in he ange θc<θ ≤ 37°, he ze o o D(N) p oduces a e y sha p minimum in he complex plane o such an ex en ha , al hough i occu s o ypical alues o Nim ~ lO-4, i s exis ence is no e ealed in he axis Ni = 0. In hese condi ions, in he plo equi alen o he one shown in Fig. 2 o (N ), he co ec minimum does no appea , leading o he ailu e o he basic assump ion o Fig. 2. Decimal loga i hm o ƒ as a unc ion o N o he inhomoge- neous wa eguide wi h θ = 80°. N m = 2.20805 co esponds o he ac ual oo o he unc ion D{N) o he leaky mode; o he pu e guided mode N m =Nm = 2.28942. Fig. 3. Decimal loga i hm o ƒ1 as a unc ion o N o he inhomoge- neous wa eguide wi h θ = 20°: a, N0im = 0; b, N0im„ = 10-4; c, N0im = 1.3 X 10-4. he LPM. This beha io can be clea ly seen in Fig. 3 o θ = 20°. Howe e , in he abo e men ioned ha m ul ange o θ alues, he unc ion (N ) s ill shows one local minimum simila o he one appea ing in Fig. 2, which, as al eady poin ed ou , does no co espond o any ze o o D (N). This local minimum misleads he minimiza ion algo i hm and leads o he e oneous alues o N m shown in Table I. The ob ained alues o Nim in hese condi ions make no sense, so hey a e no included in he able. In conclusion, ou esul s show ha he LPM p o ides e y accu a e esul s in a ious cases wi h small compu e 20 June 1991 / Vol. 30, No. 18 / APPLIED OPTICS 2419 2420 APPLIED OPTICS / Vol. 30. No. 18 / 20 June 1991 imes. Un o una ely, i does no wo k in o he cases, when he eigen alue equa ion shows a sha ply peaked o oscilla- o y beha io in he complex plane. These esul s ha e deep implica ions because hey show ha when he LPM is ap- plied o a gi en leaky o lossy s uc u e, jus o ob ain a good ag eemen in a pa ial check o he app oxima e esul s is no enough o gua an ee he iabili y o he p ocedu e when he me hod is applied o a di e en sys em. In ac , i is no ob ious how o a p io i es ablish he condi ions a which he unc ion D(N) exhibi s he ha m ul beha io leading o he ailu e o he p ocedu e. Howe e , i seems ha he p ob- lema ic cases ollow om special si ua ions o be iden i ied, a ques ion ha can only be answe ed a e u he in es iga- ion. Finally, we ha e shown also ha when his ha m ul beha io does no occu , he LPM cons i u es a e y powe - ul compu a ional ool. This wo k was sponso ed by a g an o he CIRIT o he Ca alonia Go e nmen . The au ho s a e g a e ul o Elsa Ga mi e o he Uni e si y o Sou he n Cali o nia o alu- able sugges ions. Re e ences 1. M. R. Ramadas, E. M. Ga mi e, A. K. Gha ak, K. Thyaga ajan, and M. R. Shenoy, "Analysis o Abso bing and Leaky Plana Wa eguides: a No el Me hod," Op . Le . 14, 376-378 (1989). 2. L. To ne , F. Canal, and J. He nandez-Ma co, "Leaky Modes in Mul ilaye Uniaxial Op ical Wa eguides," Appl. Op . 29, 2805- 2814 (1990). 3. J. C y oky and M. Cada, "Guided and Semileaky Modes in Aniso- opic Wa eguides o he LiNbO3 Type," Op . Commun. 27, 353- 357 (1978).