Abstract

We present a new approach to the computation of an electrical field propagating in a dielectric structure. We use the Green’s-function technique to compute an exact solution of the wave equation. No paraxial approximation is made, and our method can handle any kind of dielectric medium (air, semiconductor, metal, etc.). An original iterative numerical scheme based on the parallel use of Lippman–Schwinger and Dyson’s equations is demonstrated. The influence of the numerical parameters on the accuracy of the results is studied in detail, and the high precision and stability of the method are assessed. Examples for one and two dimensions establish the versatility of the method and its ability to handle structures of arbitrary shape. The application of the method to the computation of eigenmode spectra for dielectric structures is illustrated.

© 1994 Optical Society of America

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. M. D. Feit, J. A. Fleck, “Calculation of dispersion in graded-index multimode fibers by a propagating-beam method,” Appl. Opt. 18, 2843–2851 (1979).
    [CrossRef] [PubMed]
  3. M. D. Feit, J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]
  4. J. van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,”J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  5. L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1982).
    [CrossRef]
  6. D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
    [CrossRef]
  7. Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
    [CrossRef]
  8. T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
    [CrossRef]
  9. M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. 73, 41–47 (1990).
  10. J. Gerdes, R. Pregla, “Beam-propagation algorithm based on the method of lines,” J. Opt. Soc. Am. B 8, 389–394 (1991).
    [CrossRef]
  11. S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” IEEE J. Lightwave Technol. 7, 2033–2038 (1989).
    [CrossRef]
  12. D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
    [CrossRef] [PubMed]
  13. A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
    [CrossRef]
  14. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef] [PubMed]
  15. L. Thylén, C. M. Lee, “Beam-propagation method based on matrix diagonalization,” J. Opt. Soc. Am. A 9, 142–146 (1992).
    [CrossRef]
  16. M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,”IEEE Proc. J 138,185–190 (1991).
  17. W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
    [CrossRef]
  18. P. L. Liu, B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
    [CrossRef]
  19. R. Clauberg, P. von Allmen, “Vectorial beam-propagation method for integrated optics,” Electron. Lett. 27, 654–655 (1991).
    [CrossRef]
  20. W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
    [CrossRef]
  21. O. J. F. Martin, R. Clauberg, P. von Allmen, “Demonstration of the three-dimensional vectorial beam propagation,” presented at the European Conference on Optical Communication, Paris, September 9–12, 1991.
  22. W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
    [CrossRef]
  23. Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
    [CrossRef]
  24. J. M. Liu, L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 1574–1585 (1992).
    [CrossRef]
  25. E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
    [CrossRef]
  26. N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
    [CrossRef]
  27. H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
    [CrossRef]
  28. C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
    [CrossRef]
  29. A. Dereux, D. W. Pohl, “The 90° prism as a model SNOM probe: near-field, tunneling, and light scattering properties,” in Proceedings of the NATO Advanced Research Workshop on Near Field Optics, D. W. Pohl, ed. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 189–198.
    [CrossRef]
  30. D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, San Diego, Calif., 1974).
  31. E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, Berlin, 1990).
  32. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).
  33. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  34. G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).
  35. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).
  36. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).
  37. D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, R. Wiesendanger, H. J. Güntherodt, eds. (Springer-Verlag,Berlin, 1992), pp. 233–271.
    [CrossRef]
  38. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  39. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  40. G. N. Kamm, “Computer Fourier-transform techniques for precise spectrum measurements of oscillatory data with applications to the de Haas–van Alphen effect,” J. Appl. Phys. 49, 5951–5970 (1978).
    [CrossRef]
  41. C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
    [CrossRef]

1993

C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
[CrossRef]

1992

J. M. Liu, L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 1574–1585 (1992).
[CrossRef]

C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
[CrossRef]

G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
[CrossRef] [PubMed]

L. Thylén, C. M. Lee, “Beam-propagation method based on matrix diagonalization,” J. Opt. Soc. Am. A 9, 142–146 (1992).
[CrossRef]

P. L. Liu, B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
[CrossRef]

1991

Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
[CrossRef]

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
[CrossRef]

R. Clauberg, P. von Allmen, “Vectorial beam-propagation method for integrated optics,” Electron. Lett. 27, 654–655 (1991).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,”IEEE Proc. J 138,185–190 (1991).

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
[CrossRef]

J. Gerdes, R. Pregla, “Beam-propagation algorithm based on the method of lines,” J. Opt. Soc. Am. B 8, 389–394 (1991).
[CrossRef]

1990

D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
[CrossRef] [PubMed]

M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. 73, 41–47 (1990).

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
[CrossRef]

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

1989

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” IEEE J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

1982

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1982).
[CrossRef]

1981

1980

1979

1978

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[CrossRef] [PubMed]

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

G. N. Kamm, “Computer Fourier-transform techniques for precise spectrum measurements of oscillatory data with applications to the de Haas–van Alphen effect,” J. Appl. Phys. 49, 5951–5970 (1978).
[CrossRef]

1951

G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).

Baken, N. H. G.

H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
[CrossRef]

E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
[CrossRef]

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

Bastiaansen, H. J. M.

H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
[CrossRef]

Blok, H.

H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
[CrossRef]

E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
[CrossRef]

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

Bouju, X.

Chaudhuri, S. K.

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” IEEE J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

Chu, S. T.

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
[CrossRef]

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” IEEE J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

Chung, Y.

Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
[CrossRef]

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Clauberg, R.

R. Clauberg, P. von Allmen, “Vectorial beam-propagation method for integrated optics,” Electron. Lett. 27, 654–655 (1991).
[CrossRef]

O. J. F. Martin, R. Clauberg, P. von Allmen, “Demonstration of the three-dimensional vectorial beam propagation,” presented at the European Conference on Optical Communication, Paris, September 9–12, 1991.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Dagli, N.

Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
[CrossRef]

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

Davies, J. B.

T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Dereux, A.

C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
[CrossRef]

A. Dereux, D. W. Pohl, “The 90° prism as a model SNOM probe: near-field, tunneling, and light scattering properties,” in Proceedings of the NATO Advanced Research Workshop on Near Field Optics, D. W. Pohl, ed. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 189–198.
[CrossRef]

Diemeer, M. B. J.

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Economou, E. N.

E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, Berlin, 1990).

Feit, M. D.

Fleck, J. A.

Gerdes, J.

Girard, C.

C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
[CrossRef]

C. Girard, X. Bouju, “Self-consistent study of dynamical and polarization effects in near-field optical microscopy,” J. Opt. Soc. Am. B 9, 298–305 (1992).
[CrossRef]

Glasner, M.

Gomelsky, L.

Hadley, G. R.

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

Huang, W.

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Huang, W. P.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
[CrossRef]

Hyman, M. A.

G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kamm, G. N.

G. N. Kamm, “Computer Fourier-transform techniques for precise spectrum measurements of oscillatory data with applications to the de Haas–van Alphen effect,” J. Appl. Phys. 49, 5951–5970 (1978).
[CrossRef]

Kaplan, S.

G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).

Koch, T. B.

T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Kolk, E. W.

E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
[CrossRef]

Lagasse, P. E.

Laloe, F.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

Lee, C. M.

Li, B. J.

P. L. Liu, B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

Liu, J. M.

Liu, P. L.

P. L. Liu, B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

Majd, M.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, San Diego, Calif., 1974).

Martin, O. J. F.

C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
[CrossRef]

O. J. F. Martin, R. Clauberg, P. von Allmen, “Demonstration of the three-dimensional vectorial beam propagation,” presented at the European Conference on Optical Communication, Paris, September 9–12, 1991.

Matsuhara, M.

M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. 73, 41–47 (1990).

O’Brien, G. G.

G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).

Petermann, K.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Pohl, D. W.

D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, R. Wiesendanger, H. J. Güntherodt, eds. (Springer-Verlag,Berlin, 1992), pp. 233–271.
[CrossRef]

A. Dereux, D. W. Pohl, “The 90° prism as a model SNOM probe: near-field, tunneling, and light scattering properties,” in Proceedings of the NATO Advanced Research Workshop on Near Field Optics, D. W. Pohl, ed. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 189–198.
[CrossRef]

Pregla, R.

Splett, A.

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
[CrossRef]

Stern, M. S.

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,”IEEE Proc. J 138,185–190 (1991).

Thylén, L.

L. Thylén, C. M. Lee, “Beam-propagation method based on matrix diagonalization,” J. Opt. Soc. Am. A 9, 142–146 (1992).
[CrossRef]

Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
[CrossRef]

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1982).
[CrossRef]

van der Donk, J.

van Roey, J.

Van Splunter, J. M.

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

von Allmen, P.

R. Clauberg, P. von Allmen, “Vectorial beam-propagation method for integrated optics,” Electron. Lett. 27, 654–655 (1991).
[CrossRef]

O. J. F. Martin, R. Clauberg, P. von Allmen, “Demonstration of the three-dimensional vectorial beam propagation,” presented at the European Conference on Optical Communication, Paris, September 9–12, 1991.

Wickramasinghe, D.

T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

Xu, C.

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

Xu, C. L.

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Yevick, D.

D. Yevick, M. Glasner, “Forward wide-angle light propagation in semiconductor rib waveguides,” Opt. Lett. 15, 174–176 (1990).
[CrossRef] [PubMed]

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

Appl. Opt.

Electron. Commun. Jpn.

M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. 73, 41–47 (1990).

Electron. Lett.

D. Yevick, B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461–462 (1989).
[CrossRef]

T. B. Koch, J. B. Davies, D. Wickramasinghe, “Finite element/finite difference propagation algorithm for integrated optics,” Electron. Lett. 25, 514–516 (1989).
[CrossRef]

Y. Chung, N. Dagli, L. Thylén, “Explicit finite difference vectorial beam propagation method,” Electron. Lett. 27, 2119–2120 (1991).
[CrossRef]

R. Clauberg, P. von Allmen, “Vectorial beam-propagation method for integrated optics,” Electron. Lett. 27, 654–655 (1991).
[CrossRef]

IEEE J. Lightwave Technol.

W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” IEEE J. Lightwave Technol. 10, 295–305 (1992).
[CrossRef]

S. T. Chu, S. K. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided-wave optical structures,” IEEE J. Lightwave Technol. 7, 2033–2038 (1989).
[CrossRef]

N. H. G. Baken, M. B. J. Diemeer, J. M. Van Splunter, H. Blok, “Computational modeling of diffused channel waveguides using a domain integral equation,” IEEE J. Lightwave Technol. 8, 576–586 (1990).
[CrossRef]

IEEE J. Quantum Electron.

Y. Chung, N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1339 (1990).
[CrossRef]

P. L. Liu, B. J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. 28, 778–782 (1992).
[CrossRef]

IEEE Photon. Technol. Lett.

W. P. Huang, S. T. Chu, S. K. Chaudhuri, “A semivectorial finite-difference time-domain method,”IEEE Photon. Technol. Lett. 3, 803–806 (1991).
[CrossRef]

A. Splett, M. Majd, K. Petermann, “A novel beam propagation method for large refractive index steps and large propagation distances,”IEEE Photon. Technol. Lett. 3, 466–468 (1991).
[CrossRef]

W. P. Huang, C. L. Xu, S. T. Chu, S. K. Chaudhuri, “A vector beam propagation method for guided-wave optics,”IEEE Photon. Technol. Lett. 3, 910–913 (1991).
[CrossRef]

IEEE Proc. J

M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,”IEEE Proc. J 138,185–190 (1991).

IEEE Trans. Microwave Theory Tech.

E. W. Kolk, N. H. G. Baken, H. Blok, “Domain integral equation analysis of integrated optical channel and ridge waveguides in stratified media,” IEEE Trans. Microwave Theory Tech. 38, 78–85 (1990).
[CrossRef]

H. J. M. Bastiaansen, N. H. G. Baken, H. Blok, “Domain integral analysis of channel waveguides in anisotropic multi layered media,” IEEE Trans. Microwave Theory Tech. 40, 1918–1926 (1992).
[CrossRef]

J. Appl. Phys.

G. N. Kamm, “Computer Fourier-transform techniques for precise spectrum measurements of oscillatory data with applications to the de Haas–van Alphen effect,” J. Appl. Phys. 49, 5951–5970 (1978).
[CrossRef]

J. Math. Phys.

G. G. O’Brien, M. A. Hyman, S. Kaplan, “A study of the numerical solution of partial differential equations,”J. Math. Phys. 29, 223–251 (1951).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

L. Thylén, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1982).
[CrossRef]

Proc. IEEE

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Surf. Sci.

C. Girard, A. Dereux, O. J. F. Martin, “Field susceptibility of a composite system: application to van der Waals dispersive interactions inside a finite line of physisorbed atoms,” Surf. Sci. 295, 445–456 (1993).
[CrossRef]

Other

O. J. F. Martin, R. Clauberg, P. von Allmen, “Demonstration of the three-dimensional vectorial beam propagation,” presented at the European Conference on Optical Communication, Paris, September 9–12, 1991.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, Paris, 1977).

D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, R. Wiesendanger, H. J. Güntherodt, eds. (Springer-Verlag,Berlin, 1992), pp. 233–271.
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

A. Dereux, D. W. Pohl, “The 90° prism as a model SNOM probe: near-field, tunneling, and light scattering properties,” in Proceedings of the NATO Advanced Research Workshop on Near Field Optics, D. W. Pohl, ed. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 189–198.
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, San Diego, Calif., 1974).

E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, Berlin, 1990).

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).

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Figures (11)

Fig. 1
Fig. 1

Geometry used in the model. The dielectric medium for which we seek a solution of the wave equation can be split into a homogeneous reference medium εref and an arbitrary perturbation εp embedded in the reference medium. The discretization grid is also shown.

Fig. 2
Fig. 2

Plane wave impinging upon a dielectric barrier. The reference medium εref corresponds to the fundamental level, whereas the perturbation εp describes the barrier.

Fig. 3
Fig. 3

Amplitude of the computed field corresponding to the geometry depicted in Fig. 2, normalized to the incident field. Three dielectric barriers εp with different absorption values are investigated. The reference medium is a vacuum, and the wavelength of the incoming wave is λ0 = 0.8 μm The mesh size for the calculation was Δz = 0.01 μm.

Fig. 4
Fig. 4

Norm of the difference between numerical and analytical results as a function of the mesh size Δz. The corresponding geometry is depicted in Fig. 2. Three barriers of 1-μm thickness and varying dielectric constant εp are investigated. The wavelength in a vacuum is 0.8 μm.

Fig. 5
Fig. 5

Norm of the difference between numerical and analytical results as a function of the mesh size Δz for different effective wavelengths in the barrier depicted in Fig. 2.

Fig. 6
Fig. 6

Interaction of a plane wave in a vacuum (εeff = 1) (a) with a dielectric pad (εp = 2) and (b) with a gold pad (εp = −42.8 + i1.3). The spacing between the isoamplitude curves is 10% of the corresponding maximum–minimum amplitude range. The wavelength of the incoming wave is λ0 = 1 μm, and the mesh size is Δx = Δz = 0.05 μm.

Fig. 7
Fig. 7

Interaction of a plane wave in a vacuum (εref = 1) with a gold plate (thickness 0.1 μm, εp = −42.8 + i1.3) oriented at a 45° angle to the direction of the incoming wave. The spacing between the isoamplitude curves is 10% of the maximum–minimum amplitude range. The wavelength is λ0 = 1 μm, and the mesh size is Δx = Δz = 0.05 μm.

Fig. 8
Fig. 8

Representation of the time-averaged Poynting vector 〈S〉 corresponding (a) to Fig. 6(a) and (b) to Fig. 7. The orientation and the size of the arrows give the direction and the relative amplitude of 〈S〉, respectively.

Fig. 9
Fig. 9

Field computed for a monomode slab waveguide with a 1-μm core of dielectric constant εp = 1.1 and of claddings εref = 1. The wavelength is λ0 = 1 μm, and the mesh size is Δx = Δz = 0.05 μm. The spacing between the isoamplitude curves is 10% of the maximum–minimum amplitude range.

Fig. 10
Fig. 10

Mode spectrum of a slab waveguide with a 0.5-μm core of dielectric constant εp = 2.25 and of claddings εref = 1 for a wavelength of 1 μm, For this calculation the computational window extended only over the width of the core and was 25 μm long in the z direction; the mesh size was Δx = 0.025 μm, and Δz= 0.2 μm.

Fig. 11
Fig. 11

Error of the computed wave number β relative to its theoretical value βtheo as a function of the mesh size.

Equations (15)

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2 ψ ( r ) + ω 2 μ 0 ɛ 0 ɛ ( r ) ψ ( r ) = 0
ɛ ( r ) = ɛ ref + Δ ɛ p ( r ) ,
ψ ( r ) = ψ 0 ( r ) + perturbation d r G 0 ( r , r ) V ( r ) ψ ( r ) ,
V ( r ) = k 0 2 Δ ɛ p ( r ) ,
k 0 2 = ω 2 ɛ 0 μ 0 .
ψ i = ψ i 0 + k = 1 N p G i , k 0 V k Δ k ψ k .
G ( r , r ) = G 0 ( r , r ) + perturbation d r G 0 ( r , r ) V ( r ) G ( r , r )
G i , j = G i , j 0 + k = 1 N p G i , k 0 V k Δ k G k , j .
ψ i 1 = ψ i 0 + G i , k 1 0 V k 1 Δ k 1 ψ k 1 ,
G i , j 1 = G i , j 0 + G i , k 1 0 V k 1 Δ k 1 G k 1 , j 1 .
ψ i 2 = ψ i 1 + G i , k 2 1 V k 2 Δ k 2 ψ k 2 ,
G i , j 2 = G i , j 1 + G i , k 2 1 V k 2 Δ k 2 G k 2 , j 2 .
ψ 0 ( z ) = exp ( i k ref z )
ψ ϕ 2 = 0 L d z [ ϕ ( z ) ψ ( z ) ] * [ ϕ ( z ) ψ ( z ) ] .
η ( z ) = ω ω d x ψ ( x , z ) .

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