1134 JOURNAL OF LIGHTWAVE TECHNOLOGY.VOL.6.NO.6.JUNE 1988 percent),and the method predicts the position and the 16]M.Fukuma.J.Noda,and H.Iwasaki.Optical properties of tita value of the field maximum very accurately.The discrep- nium-diffused LiNbO,strip waveguides,'J.Appl.Phys..vol.49. ancies arising in the lateral field distribution X(x)are pp.3693-3698.1978 17]G.P.Bava.I.Montrosset,W.Sohler,and H.Suche,'Numerical higher and usually a wider spot size is obtained.In fact modeling of Ti:LiNbO,integrated optical parametric oscillators." the solution of (8b)deals with a refractive index distri IEEE J.Quantum Electron..vol.QE-23,pp.42-51,1987. 8]M.V.Hobden and J.Warner.The temperature dependence of the bution n(s)becoming constant and equal to ng for s2 refractive indices of pure lithium niobate.Phys.Ler.vol.22.pp. sm while the actual index profile approaches n for s 243-244.1966. [9]H.Luidtke,W.Sohler,and H.Suche,"Characterization of Therefore the simulated waveguide used in the ERI Ti:LiNbO,optical waveguides,in Dig.Workshop Integrated Op- method has a smaller width than the original waveguide tics."R.Th.Kersten and R.Ulrich.Eds. Berlin,W.Germany as used by FEM and the lateral confinement is reduced. Technical Univ.Berlin,1980,pp.122-126. The choice of M and R values different from unity does 110]M.Minakata,S.Saito,and M.Shibata.'Two-dimensional distri bution of refractive index changes in Ti-diffused LiNbO,strip wave- not overcome this problem but allows better results for guides,J.Appl.Phys..vol.50.pp.3063-3067,1979 the normalized propagation constant b in the range of pa- [11]S.Fouchet,R.Guglielmi.A.Yi-Yan.and A.Carenco,Wavelength rameters where the ERI method is valid,i.e.,in case an dispersion of Ti:LiNbO:waveguides.'in Proc.2nd Eur.Conf.In- tegrated Optics (Florence.Italy).1983.pp.50-52. accurate approximation of the refractive index profile [12]G.Arvidsson and F.Laurell.Nonlinear wavelength conversion in mainly near the waveguide axis is required. Ti:LiNbO,waveguides,'Thin Solid Films.vol.136.p.29.1986. 113]T.Suhara.Y.Handa,H.Nishihara.J.Koyama,'Analysis of optical VI.CONCLUSION channel waveguides and directional couplers with graded-index pro file.''J.Opt.Soc.Amner.,vol.69,pp.807-815,1979. In this paper two methods for the calculation of optical [14]M.Koshiba.K.Hayata.and M.Suzuki.Approximatc scalar finite element analysis of anisotropic optical waveguides with off-diagonal modes of Ti:LiNbO:channel waveguides have been pre- elements in a permittivity tensor.IEEE Trans.Microwave Theory sented.The results computed using the purely numerical Tech.vol.MTT-32.pp.587-593.1984. finite-element method and the nearly analytical effective- 15]G.B.Hocker and W.K.Burns.Mode dispersion in diffused channel refractive-index method have been compared.The ap- waveguides by the effective index method,Appl.Opr.,vol.16.pp. 113-118.1977. proximations used in the ERI method lead to analytical 116]W.Streifer,D.R.Scifres,and R.D.Burnham.''Analysis of gain- expressions for the field distributions and effective in- induced waveguiding in stripe geometry diode lasers.'/EEE J. dices.The accuracy is generally good for well guided Quantum Electron..vol.QE-14.pp.418-427.1978. 117]H.A.Haus and R.V.Schmidt.''Approximate analysis of optical modes.The propagation constant and the position of the waveguide grating coupling coetficients.'Appl.Opt..vol.15,pp. fundamental mode field maximum are very accurate also 774-781.1976. for modes near cutoff,whereas the field distributions.es- [18]M.Abramovitz and I.Stegun,Handbook of Mathemarical Functions New York:Dover,1972,ch.22. pecially in the lateral (width)direction,become less ac- 119]D.Marcuse,Theory of Dielectric Optical Waveguides. New York curate for weakly guided modes.The proposed index pro- Academic 974 file approximation in the ERI method therefore leads to a 120]K.Hayata.H.Koshiba.M.Eguchi.and M.Suzuki."'Novel finite- clement formulation without any spurious solutions for dielectric very fast technique for those applications where a large waveguides.Electron.Lett..vol.22.pp.295-296.1986. number of computations have to be performed with the 121]E.Strake,"Berechnung und Messung von Feldverteilungen optischer Moden in dielektrischen Wellenleitern'',diploma thesis.Universitat- main interest lying on data like propagation constant, GH-Paderborn,Paderborn.W.Germany,1984. mode size or position of the mode field maximum.This [22]K.Hayata,M.Koshiba,and M.Suzuki,"Lateral mode analysis of case typically arises in the design and optimization of in- buried heterostructure diode lasers by the finite-element method.' tegrated optical devices. IEEE J.Quantum Electron..vol.QE-22.pp.781-788.1986. [23]H.R.Schwarz.Methode der finiten Elemente. Stuttgart.W.Ger- In all problems requiring an exact numerical solution of many:Teubner,1980. the wave equation while computation time is unimportant [24]S.K.Korotky,W.J.Minford.L.L.Buhl.M.D.Divino.and R.C. a numerical technique like the finite-element method must Alferness."Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO:strip waveguides EEE be used.Its main advantage besides its high accuracy is Quantum Electron..vol.QE-18,pp.1796-1801,1982. that it can be applied to waveguides with arbitrary refrac- tive index profiles. Engelbert Strake was bor in Neuenkirchen/Westfalen.W.Germany,in REFERENCES 1958.He received the M.S.degree in physics from the University of Pa- derborn.Paderborn.W.Germany.in 1984. [1]R.A.Steinberg and T.G.Giallorenzi.''Modal fields of anisotropic Since 1986 he has been working for the Ph.D.degree at the University channel waveguides,"J.Opt.Soc.Amer..vol.67.pp.523-533. of Paderborn.Department of Applied Physics.in the field of integrated 1977. optics 12]C.Yeh,K.Ha,S.B.Dong,and W.P.Brown.'Single-mode optical waveguides.App/.Opr..vol.18.pp.1490-1504.1979. 13]N.Mabaya,P.E.Lagasse.and P.Vandenbulcke,Finite element analysis of optical waveguides,'IEEE Trans.Microwave Theory Gian Paolo Bava was born in Varallo,Italy,in 1937.He received the Trh.vol.MTT-29,pp.600-605.1981. degrec in electrical engineering from Politecnico di Torino.Torino.Italy. 14]J.-D.Decotignie,O.Parriaux.and F.E.Gardiol,Wave propaga in1961, tion in an inhomogeneous diffused channel guide using a finite-differ. Since 1961 he has been with the Institute of Electrical Communications ence technique,Proc.Ist Eur.Conf.Integrated Opt.(London,En- (succesively Department of Electronics),Politecnico di Torino.From 1970 gland).1981.Pp.40-42, to 1976 he was in charge of the course on microwave techniques at the 15]W.K.Burns,D.H.Klcin.and E.J.West.Ti diffusion in Department of Electronics.At present his main research activities are con- Ti:LiNbO,planar and channel optical waveguides.''J.Appl.Phys.. cerned with microwave MESFET amplifiers and oscillators and integrated vol.50,pp.6175-6182.1979 optics.1134 JOURNAL OF LIGHTWAVE TECHNOLOGY. VOL. 6. NO 6, JUNE 1988 percent), and the method predicts the position and the value of the field maximum very accurately. The discrep￾ancies arising in the lateral field distribution X(x) are higher and usually a wider spot size is obtained. In fact the solution of (8b) deals with a refractive index distri￾bution n&( s) becoming constant and equal to ni for s I s,,~ while the actual index profile approaches nb for s -+ W. Therefore the simulated waveguide used in the ER1 method has a smaller width than the original waveguide as used by FEM and the lateral confinement is reduced. The choice of M and R values different from unity does not overcome this problem but allows better results for the normalized propagation constant b in the range of pa￾rameters where the ER1 method is valid, i.e., in case an accurate approximation of the refractive index profile mainly near the waveguide axis is required. VI. CONCLUSION In this paper two methods for the calculation of optical modes of Ti : LiNbO, channel waveguides have been pre￾sented. The results computed using the purely numerical finite-element method and the nearly analytical effective￾refractive-index method have been compared. The ap￾proximations used in the ER1 method lead to analytical expressions for the field distributions and effective in￾dices. The accuracy is generally good for well guided modes. The propagation constant and the position of the fundamental mode field maximum are very accurate also for modes near cutoff, whereas the field distributions, es￾pecially in the lateral (width) direction, become less ac￾curate for weakly guided modes. The proposed index pro￾file approximation in the ER1 method therefore leads to a very fast technique for those applications where a large number of computations have to be performed with the main interest lying on data like propagation constant, mode size or position of the mode field maximum. This case typically arises in the design and optimization of in￾tegrated optical devices. In all problems requiring an exact numerical solution of the wave equation while computation time is unimportant a numerical technique like the finite-element method must be used. Its main advantage besides its high accuracy is that it can be applied to waveguides with arbitrary refrac￾tive index profiles. [I1 121 131 141 151 REFERENCES R. A. Steinberg and T. G. Giallorenzi. “Modal fields of anisotropic channel waveguides,” J. Opt. Soc. Amer., vol. 67, pp. 523-533, 1977. C. Yeh, K. Ha, S. B. Dong, and W. P. Brown, “Single-mode optical waveguides,” Appl. Opt., vol. 18, pp. 1490-1504, 1979. N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” fEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 600-605, 1981. J:D. Decotignie, 0. Parriaux, and F. E. Gardiol, “Wave propaga￾tion in an inhomogeneous diffused channel guide using a finite-differ￾ence technique,” Proc. 1st Eur. Con$ fntegrated Opt. (London, En￾gland), 1981, pp. 40-42. W. K. Burns, D. H. Klein, and E. J. West, “Ti diffusion in Ti : LiNbO, planar and channel optical waveguides,” J. Appl. Phys., vol. 50, pp. 617556182. 1979. M. Fukuma, J. Noda. and H. lwasaki, “Optical properties of tita￾nium-diffused LiNbO, strip waveguides.” 1. Appl. Phys.. vol. 49, pp. 3693-3698, 1978. G. P. Bava, I. Montrosset. W. Sohler. and H. Suche, “Numerical modeling of Ti : LiNbO, integrated optical parametric oscillators,” fEEEJ. Quantum Electron.. vol. QE-23, pp. 42-51, 1987. M. V. Hobden and J. Warner, “The temperature dependence of the refractive indices of pure lithium niobate,” Phvs. Lett., vol. 22, pp. H. Ludtke, W. Sohler, and H. Suche, “Characterization of Ti : LiNbO? optical waveguides,” in Dig. Workshop Integrared Op￾tics,” R. Th. Kersten and R. Ulrich, Eds. Berlin, W. Germany: Technical Univ. Berlin, 1980, pp. 122-126. M. Minakata, S. Saito, and M. Shibata, “Two-dimensional distri￾bution of refractive index changes in Ti-diffused LiNbO, strip wave￾guides,” .I. Appl. Phys., vol. 50, pp. 3063-3067, 1979. S. Fouchet, R. Guglielmi, A. Yi-Yan. and A. Carenco, “Wavelength dispersion of Ti:LiNbO, waveguides,” in Proc. 2nd Eur. Con$ In￾tegrated Optics (Florence, Italy), 1983, pp. 50-52. G. Arvidsson and F. Laurell. “Nonlinear wavelength conversion in Ti: LiNbO, waveguides,” Thin Solid Films, vol. 136. p. 29, 1986. T. Suhara, Y. Handa, H. Nishihara, J. Koyama, “Analysis of optical channel waveguides and directional couplers with graded-index pro￾file.” J. Opt. Soc. Amer., vol. 69, pp, 8077815, 1979. M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite￾element analysis of anisotropic optical waveguides with off-diagonal elements in a permittivity tensor,“ fEEE Truris. Microwave Theory Tech.. vol. MTT-32. pp. 587-593, 1984. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt., vol. 16, pp. W. Streifer, D. R. Scifres. and R. D. Burnham. “Analysis of gain￾induced waveguiding in stripe geometry diode lasers.” fEEE J. Quuntum ELectron., vol. QE-14, pp. 418-427, 1978. H. A. Haus and R. V. Schmidt, “Approximate analysis of optical waveguide grating coupling coefficients,” Appl. Opt., vol. 15, pp. M. Abrdmovitz and I. Stegun, Hundhook ofMuthc~n~aticul Functions. New York: Dover, 1972, ch. 22. D. Marcuse, Theory of Dielectric OpticNI Wuvrguidrs. New York: Academic, 1974. K. Hayata, H. Koshiba. M. Eguchi. and M. Suzuki, “Novel finite￾element formulation without any spurious solutions for dielectric waveguides,” Electron. Letr., vol. 22, pp. 295-296, 1986. E. Strake, “Berechnung und Messung von Feldverteilungen optischer Moden in dielektrischen Wellenleitern”. diploma thesis, Universitit￾GH-Paderborn, Paderborn, W. Germany, 1984. K. Hayata. M. Koshiba, and M. Suzuki, “Lateral mode analysis of buried heterostructure diode lasers by the finite-element method.” IEEE J. Quanrum Electrori.. vol. QE-22, pp. 781-788, 1986. H. R. Schwarz. Merhork drrjitiircw Elemente. Stuttgart. W. Ger￾many: Teubner, 1980. S. K. Korotky, W. J. Minford. L. L. Buhl, M. D. Divino, and R. C. Alferness. “Mode size and method for estimating the propagation 243-244, 1966. 113-118, 1977. 774-781, 1976. constant of single-mode TI : LiNbO? strip waveguides,.” iEEE J. Qucnitum Electron., vol. QE- 18, pp. 1796- I 80 1, 1982. * Engelbert Strake was born in NeuenkircheniWestfalen, W. Germany, in 1958. He received the M.S. degree in physics from the University of Pa￾derborn, Paderborn, W. Germany, in 1984. Since 1986 he has been working for the Ph.D. degree at the University of Paderborn, Department of Applied Physics, in the field of integrated optics. * Gian Paulo Bava was born in Varallo, Italy, in 1937. He received the degree in electrical engineering from Politecnico di Torino, Torino, Italy, in 1961. Since 1961 he has been with the lnstitute of Electrical Communications (succesively Department of Electronics), Politecnico di Torino. From 1970 to 1976 he was in charge of the course on microwave techniques at the Department of Electronics. At present his main research activities are con￾cerned with microwave MESFET amplifiers and oscillators and integrated optics