Computing guided modes for an unbounded stratified medium in integrated optics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, p. 799-824

We present a finite element method to compute guided modes in a stratified medium. The major difficulty to overcome is related to the unboundedness of the stratified medium. Our method is an alternative to the use of artificial boundary conditions and to the use of integral representation formulae. The domain is bounded in such a way we can write the solution on its lateral boundaries in terms of Fourier series. The series is then truncated for the computations over the bounded domain. The problem is scalar and 2-dimensional.

Classification:  65N30,  65N25,  35Q60,  78M10,  68U20
Keywords: finite element method, exact boundary condition, unbounded domain, stratified medium, guided modes, optics, series expansion
@article{M2AN_2001__35_4_799_0,
     author = {Mah\'e, Fabrice},
     title = {Computing guided modes for an unbounded stratified medium in integrated optics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     pages = {799-824},
     zbl = {0993.78017},
     mrnumber = {1863281},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_4_799_0}
}
Mahé, Fabrice. Computing guided modes for an unbounded stratified medium in integrated optics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 4, pp. 799-824. http://www.numdam.org/item/M2AN_2001__35_4_799_0/

[1] A.S. Bonnet, Analyse mathématique de la propagation de modes guidés dans les fibres optiques. Ph.D. thesis, University of Paris VI (1988).

[2] A.S. Bonnet-Bendhia, G. Caloz, M. Dauge and F. Mahé, Study at high frequencies of a stratified waveguide. IMA J. Appl. Math. 66 (2001) 231-257. | Zbl 0986.78014

[3] A.S. Bonnet-Bendhia, G. Caloz and F. Mahé, Guided modes of integrated optical guides. A mathematical study. IMA J. Appl. Math. 60 (1998) 225-261. | Zbl 0914.35130

[4] A.S. Bonnet-Bendhia and N. Gmati, Spectral approximation of a boundary condition for an eigenvalue problem. SIAM J. Numer. Anal. 32 (1995) 1263-1279. | Zbl 0834.65104

[5] A.S. Bonnet-Bendhia and P. Joly, Mathematical analysis of guided water waves. SIAM J. Appl. Math. 53 (1993) 1507-1550. | Zbl 0787.76007

[6] A.S. Bonnet-Bendhia and F. Mahé, A guided mode in the range of the radiation modes for a rib waveguide. J. Optics 28 (1997) 41-43.

[7] N. Gmati, Guidage et diffraction d'ondes en milieu non borné. Ph.D. thesis, University of Paris VI (1992).

[8] A. Jami and M. Lenoir, A variational formulation for exterior problems in linear hydrodynamics. Comput. Methods. Appl. Mech. Engrg. 16 (1978) 314-359. | Zbl 0392.76020

[9] M. Koshiba, Optical waveguide theory by the finite element method. KTC Scientific Publishers, Tokyo (1992).

[10] M. Lenoir and A. Tounsi, The localized finite element method and its applications to the two-dimensional sea-keeping problem. SIAM J. Numer. Anal. 25 (1988) 729-752. | Zbl 0656.76008

[11] F. Mahé, Étude mathématique et numérique de la propagation d'ondes électromagnétiques dans les microguides de l'optique intégrée. Ph.D. thesis, University of Rennes I, France (1993).

[12] D. Martin, Guide d'utilisation du code Mélina, IRMAR, University of Rennes I, France (1997). e-mail: http://www.maths.univ-rennes1.fr/ ˜dmartin

[13] B.M.A. Rahman and J.B. Davies, Finite-element analysis of optical and microwave waveguide problems. IEEE Trans. Microwave Theory Tech. MTT-32(1) (1984).

[14] M. Reed and B. Simon, Analysis of Operators. IV: Analysis of operators. Academic Press, New York, San Francisco, London (1978). | MR 493421 | Zbl 0401.47001

[15] A.W. Snyder and J.D. Love, Optical waveguide theory. Chapman and Hall, London (1983).

[16] M. Schechter, Spectra of partial differential operators. North-Holland, Amsterdam (1971). | MR 869254 | Zbl 0225.35001

[17] C. Vassalo, Théorie des guides d'ondes électromagnétiques. Tomes 1 et 2. Eyrolles Éditions and cnet-enst, Paris (1985).

[18] J.H. Wilkinson, The Algebraic Eigenvalue Problem. Clarenton Press, Oxford (1965). | MR 184422 | Zbl 0258.65037