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IEEE Journal of Lightwave Technology
Volume 18 Number 5, May 2000

Table of Contents for this issue

Complete paper in PDF format

Norm-Conserving Finite-Difference Beam-Propagation Method for TM Wave Analysis in Step-Index Optical Waveguides

Junji Yamauchi, Member, IEEE, Member, OSA Kenji Matsubara, Takeshi Tsuda and Hisamatsu Nakano Fellow, IEEE

Page 721.

Abstract:

Nonconservation of power is a perplexing problem in the propagating beam analysis of transverse magnetic (TM) waves in a z -variant step-index optical waveguide. To conserve the power in terms of a squared norm, a modified finite-difference (FD) formula is introduced that allows a general position of a core-cladding interface. The use of the modified formula contributes to a reduction in a field profile error caused by a staircase approximation with subsequent conservation of power, particularly for a symmetrical waveguide. To obtain the power conservation even in the analysis of an asymmetrical waveguide, a z-derivative of the refractive index is taken into account. An asymmetrical taper and tilted waveguides placed in parallel are investigated to validate the present technique.

References

  1. C. Vassallo, "Difficulty with vectorial BPM", Electron. Lett., vol. 33, pp.  61-62, 1997.
  2. H. J. W. M. Hoekstra, "On beam propagation methods for modeling in integrated optics", Opt. Quantum Electron., vol. 29, pp.  157-171, 1997.
  3. L. Poladian and F. Ladouceur, "Unification of TE and TM beam propagation algorithms", IEEE Photon. Technol. Lett., vol. 10, pp.  105 -107, 1998.
  4. C. Vassallo, "Interest of improved three-point formulas for finite-difference modeling of optical devices", J. Opt. Soc. Amer. A, vol. 14, pp.  3273-3284, 1997.
  5. F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation", J. Lightwave Technol., vol. 11, pp.  1425-1434, 1993.
  6. J. Haes, R. Baets, C. M. Weinert, M. Gravert, H. P. Nolting, M. A. Andrade, A. Leite, H. K. Bissessur, J. B. Davies, R. D. Ettinger, J. Ctyroky, E. Ducloux, F. Ratovelnomanana, N. Vodjdani, S. Helfert, R. Pregla, F. H. G. M. Wijnands, H. J. W. M. Hoekstra and G. J. M. Krijnen, "A comparison between different propagative schemes for the simulation of tapered step index slab waveguides", J. Lightwave Technol., vol. 14, pp.  1557-1569, 1996.
  7. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media", IEEE Trans. Antennas Propagat., vol. AP-14, pp.  302-307, 1966.
  8. T. Barts, J. Browman, R. K. Cooper, M. Dehler, M. Dohlus, F. Ebeling, A. Fischerauer, G. Fischerauer, P. Hahne, R. Klatt, F. Krawczyk, M. Marx, T. Pröndpper, G. Rodenz, D. Rusthoi, P. Schündtt, B. Steffer, T. Weiland and S.G. Wipf, "Maxwell's grid equations", Frequenz , vol. 44, pp.  9-19, 1990.
  9. T. Itoh, G. Pelosi, and P. P. Silvestor, Eds., Finite Element Software for Microwave Engineering, New York: Wiley, 1996.
  10. J. Yamauchi, G. Takahashi and H. Nakano, "Modified finite-difference formula for semivectorial H-field solutions of optical waveguides", IEEE Photon. Technol. Lett., vol. 10, pp.  1127-1129, 1998.
  11. J. Yamauchi, G. Takahashi and H. Nakano, "Full-vectorial beam-propagation method based on the McKee-Mitchell scheme with improved finite-difference formulas", J. Lightwave Technol., vol. 16, no. 12, pp.  2458-2464, 1998 .
  12. J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama and H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides", IEEE Photon. Technol. Lett., vol. 9, pp.  961-963, 1997.
  13. M. S. Stern, "Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles", Proc. Inst. Elect. Eng., vol. 135, no. 5, pp.  333-338, 1988 .
  14. D. Yevick and W. Bardyszewski, "Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis", Opt. Lett., vol. 17, pp.  329-330, 1992.
  15. C. L. Xu, W. P. Huang and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method", J. Lightwave Technol., vol. 11, pp.  1209-1215, 1993.
  16. S. Jüngling and J. C. Chen, "A study and optimization of eigenmode calculations using imaginary-distance beam-propagation method", IEEE J. Quantum Electron., vol. 30, pp.  2098 -2105, 1994.
  17. G. R. Hadley, "Transparent boundary condition for beam propagation", Opt. Lett., vol. 16, pp.  624-626, 1991.
  18. C. Vassallo, "Finite difference analysis of vectorial transversal fields in optical waveguides", in Proc. 3rd Int. Conf. Mathematical and Numerical Aspects of Wave Propagation (SIAM-INRIA), Mandeilieu, 1995, pp.  594-603. 
  19. Y. Tsuji, M. Koshiba and T. Tanabe, "A wide-angle beam propagation method using a finite element scheme", IEICE Trans., vol. J79-C-I, pp.  381 -388, 1996.