Abstract

The one-way wave equation in the oblique coordinate system in terms of the square root operator is derived. This equation forms the basis for the development of efficient algorithms using the beam propagation method for the design and optimization of integrated optical devices. In an illustrative example, using the derived one-way wave equation, Anada’s very-wide-angle algorithm is generalized to the oblique coordinate system. Since in the oblique coordinate system the direction of propagation can be selected freely to follow the path of the optical beam and to minimize the stair-casing errors, the algorithm is expected to show superior performance, which is confirmed by the results obtained.

© 2008 Optical Society of America

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  1. D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
    [Crossref]
  2. P. Bienstman and R. Baets, "Advanced boundary conditions for eigenmode expansion models," Opt. Quantum Electron. 34, 523-540 (2002).
    [Crossref]
  3. M. D. Feit and J. A. Fleck, Jr., "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978).
    [Crossref] [PubMed]
  4. F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation," J. Lightwave Technol. 11, 1425-1434 (1993).
    [Crossref]
  5. Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method," IEEE J. Quantum Electron. 26, 1335-1339 (1990).
    [Crossref]
  6. E. Ahlers and R. Pregla, "3-D modelling of concatenations of straight and curved waveguides by MoL-BPM," Opt. Quantum Electron. 29, 151-156 (1997).
    [Crossref]
  7. A. Splett, M. Majd, and 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]
  8. C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
    [Crossref]
  9. J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
    [Crossref]
  10. P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
    [Crossref]
  11. G. R. Hadley, "Wide-angle beam propagation using Padé approximant operators," Opt. Lett. 17, 1426-1428 (1992).
    [Crossref] [PubMed]
  12. H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992).
    [Crossref]
  13. G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992).
    [Crossref] [PubMed]
  14. T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).
  15. P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
    [Crossref]
  16. P. Sewell, T. M. Benson, S. Sujecki, and P. C. Kendall, "The dispersion characteristics of oblique coordinate beam propagation algorithms," J. Lightwave Technol. 17, 514-518 (1999).
    [Crossref]
  17. Chia-Chien Huang and Chia-Chih Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
    [Crossref]
  18. Y. Y. Lu, "A complex coefficient rational approximation of 1+x," Appl. Numer. Math. 27, 141-154 (1998).
    [Crossref]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).
  20. S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
    [Crossref]
  21. I. Ilic, R. Scarmozzino, and R. M. Osgood, "Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modelling of waveguiding circuits," J. Lightwave Technol. 14, 2813-2822 (1996).
    [Crossref]
  22. J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods," J. Lightwave Technol. 17, 677-683 (1999).
    [Crossref]
  23. C. Vassallo, "Limitations of the wide angle beam propagation method in nonuniform systems," J. Opt. Soc. Am. A 13, 761-770 (1996).
    [Crossref]

2005 (1)

Chia-Chien Huang and Chia-Chih Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
[Crossref]

2003 (1)

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

2002 (1)

P. Bienstman and R. Baets, "Advanced boundary conditions for eigenmode expansion models," Opt. Quantum Electron. 34, 523-540 (2002).
[Crossref]

1999 (3)

1998 (1)

Y. Y. Lu, "A complex coefficient rational approximation of 1+x," Appl. Numer. Math. 27, 141-154 (1998).
[Crossref]

1997 (3)

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
[Crossref]

E. Ahlers and R. Pregla, "3-D modelling of concatenations of straight and curved waveguides by MoL-BPM," Opt. Quantum Electron. 29, 151-156 (1997).
[Crossref]

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

1996 (3)

P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
[Crossref]

I. Ilic, R. Scarmozzino, and R. M. Osgood, "Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modelling of waveguiding circuits," J. Lightwave Technol. 14, 2813-2822 (1996).
[Crossref]

C. Vassallo, "Limitations of the wide angle beam propagation method in nonuniform systems," J. Opt. Soc. Am. A 13, 761-770 (1996).
[Crossref]

1995 (1)

J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
[Crossref]

1994 (1)

D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
[Crossref]

1993 (1)

F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation," J. Lightwave Technol. 11, 1425-1434 (1993).
[Crossref]

1992 (4)

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

H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992).
[Crossref]

G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992).
[Crossref] [PubMed]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).

1991 (1)

A. Splett, M. Majd, and 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]

1990 (1)

Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method," IEEE J. Quantum Electron. 26, 1335-1339 (1990).
[Crossref]

1978 (1)

Ahlers, E.

E. Ahlers and R. Pregla, "3-D modelling of concatenations of straight and curved waveguides by MoL-BPM," Opt. Quantum Electron. 29, 151-156 (1997).
[Crossref]

Anada, T.

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
[Crossref]

P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
[Crossref]

Baets, R.

P. Bienstman and R. Baets, "Advanced boundary conditions for eigenmode expansion models," Opt. Quantum Electron. 34, 523-540 (2002).
[Crossref]

Benson, T. M.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

P. Sewell, T. M. Benson, S. Sujecki, and P. C. Kendall, "The dispersion characteristics of oblique coordinate beam propagation algorithms," J. Lightwave Technol. 17, 514-518 (1999).
[Crossref]

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
[Crossref]

P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
[Crossref]

Bienstman, P.

P. Bienstman and R. Baets, "Advanced boundary conditions for eigenmode expansion models," Opt. Quantum Electron. 34, 523-540 (2002).
[Crossref]

Borruel, L.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Chung, Y.

Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method," IEEE J. Quantum Electron. 26, 1335-1339 (1990).
[Crossref]

Dagli, N.

Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method," IEEE J. Quantum Electron. 26, 1335-1339 (1990).
[Crossref]

Erbert, G.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Esquivias, I.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Fan, P.-L.

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

Feit, M. D.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).

Fleck, J. A.

Glingener, C.

D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
[Crossref]

Hadley, G. R.

Hiraoka, T.

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

Hoekstra, H. J. W. M.

H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992).
[Crossref]

Hokazono, T.

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

Hsu, J. P.

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

Hsu, J.-M.

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

Huang, Chia-Chien

Chia-Chien Huang and Chia-Chih Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
[Crossref]

Huang, Chia-Chih

Chia-Chien Huang and Chia-Chih Huang, "A novel wide-angle beam propagation method based on the spectral collocation scheme for computing tilted waveguides," IEEE Photon. Technol. Lett. 17, 1872-1874 (2005).
[Crossref]

Ilic, I.

I. Ilic, R. Scarmozzino, and R. M. Osgood, "Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modelling of waveguiding circuits," J. Lightwave Technol. 14, 2813-2822 (1996).
[Crossref]

Kendall, P. C.

P. Sewell, T. M. Benson, S. Sujecki, and P. C. Kendall, "The dispersion characteristics of oblique coordinate beam propagation algorithms," J. Lightwave Technol. 17, 514-518 (1999).
[Crossref]

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
[Crossref]

P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
[Crossref]

Krijnen, G. J. M.

H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992).
[Crossref]

Lambeck, P. V.

H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, "On the accuracy of the finite difference method for applications in beam propagating techniques," Opt. Commun. 94, 506-508 (1992).
[Crossref]

Larkins, E. C.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Lee, C.-T.

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

Lu, Y. Y.

Y. Y. Lu, "A complex coefficient rational approximation of 1+x," Appl. Numer. Math. 27, 141-154 (1998).
[Crossref]

Majd, M.

A. Splett, M. Majd, and 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]

Matsubara, K.

Moreno, P.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Nakano, H.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods," J. Lightwave Technol. 17, 677-683 (1999).
[Crossref]

J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
[Crossref]

Osgood, R. M.

I. Ilic, R. Scarmozzino, and R. M. Osgood, "Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modelling of waveguiding circuits," J. Lightwave Technol. 14, 2813-2822 (1996).
[Crossref]

Petermann, K.

A. Splett, M. Majd, and 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]

Pregla, R.

E. Ahlers and R. Pregla, "3-D modelling of concatenations of straight and curved waveguides by MoL-BPM," Opt. Quantum Electron. 29, 151-156 (1997).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).

Scarmozzino, R.

I. Ilic, R. Scarmozzino, and R. M. Osgood, "Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modelling of waveguiding circuits," J. Lightwave Technol. 14, 2813-2822 (1996).
[Crossref]

Schmidt, F.

F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation," J. Lightwave Technol. 11, 1425-1434 (1993).
[Crossref]

Schultz, D.

D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
[Crossref]

Sekiguchi, M.

Sewell, P.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

P. Sewell, T. M. Benson, S. Sujecki, and P. C. Kendall, "The dispersion characteristics of oblique coordinate beam propagation algorithms," J. Lightwave Technol. 17, 514-518 (1999).
[Crossref]

T. Anada, T. Hokazono, T. Hiraoka, J. P. Hsu, T. M. Benson, and P. Sewell, "Very-wide-angle beam propagation methods for integrated optical circuits," IEICE Trans. Electron. E82-C, 1154-1158 (1999).

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall, "Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions," J. Lightwave Technol. 15, 688-696 (1997).
[Crossref]

P. Sewell, T. Anada, T. M. Benson, and P. C. Kendall, "Non-standard beam propagation," Microwave Opt. Technol. Lett. 13, 24-26 (1996).
[Crossref]

Sheu, L.-G.

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

Shibayama, J.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods," J. Lightwave Technol. 17, 677-683 (1999).
[Crossref]

J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
[Crossref]

Splett, A.

A. Splett, M. Majd, and 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]

Sujecki, S.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

P. Sewell, T. M. Benson, S. Sujecki, and P. C. Kendall, "The dispersion characteristics of oblique coordinate beam propagation algorithms," J. Lightwave Technol. 17, 514-518 (1999).
[Crossref]

Sumpf, B.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).

Vassallo, C.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, 1992).

Voges, E.

D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
[Crossref]

Wenzel, H.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Wu, M.-L.

C.-T. Lee, M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu, "Design and analysis of completely adiabatic tapered waveguides by conformal mapping," J. Lightwave Technol. 15, 403-410 (1997).
[Crossref]

Wykes, J.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

Yamauchi, J.

J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods," J. Lightwave Technol. 17, 677-683 (1999).
[Crossref]

J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
[Crossref]

Appl. Numer. Math. (1)

Y. Y. Lu, "A complex coefficient rational approximation of 1+x," Appl. Numer. Math. 27, 141-154 (1998).
[Crossref]

Appl. Opt. (1)

Electron. Commun. Jpn., Part 2: Electron. (1)

J. Yamauchi, J. Shibayama, and H. Nakano, "Finite-difference beam propagation method using the oblique coordinate system," Electron. Commun. Jpn., Part 2: Electron. 78, 20-26 (1995).
[Crossref]

IEEE J. Quantum Electron. (2)

D. Schultz, C. Glingener, and E. Voges, "Novel generalised finite-difference beam propagation method," IEEE J. Quantum Electron. 30, 1132-1140 (1994).
[Crossref]

Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method," IEEE J. Quantum Electron. 26, 1335-1339 (1990).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, "Nonlinear properties of tapered laser cavities," IEEE J. Sel. Top. Quantum Electron. 9, 823-834 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (2)

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Figures (8)

Fig. 1
Fig. 1

Oblique coordinate system.

Fig. 2
Fig. 2

Dependence of relative error in z component of the wave vector on the propagation angle for θ = ( a ) 20°, (b) 50°, and (c) 80° in oblique coordinate system for d z = d x = 0.01 μ m .

Fig. 3
Fig. 3

Dependence of relative error in z component of the wave vector on the propagation angle for d z = d x = ( a ) 0.01, (b) 10 4 , and (c) 10 6 μ m in oblique coordinate system at θ = 50 ° .

Fig. 4
Fig. 4

Example of tilted waveguide structure.

Fig. 5
Fig. 5

Dependence of field offset on the WA approximation order for the test waveguide structure: Δ n = 0.001 for θ = ( a ) 0 and (b) 0.5 rad .

Fig. 6
Fig. 6

Dependence of field offset on the WA approximation order for the test waveguide structure: Δ n = 0.001 , d z = d x = 0.005 μ m .

Fig. 7
Fig. 7

Dependence of field offset on the mesh size for the test waveguide structure for θ = 0.5 rad : Δ n = ( a ) 0.001, (b) 0.01, (c) 0.1, and (d) 1.

Fig. 8
Fig. 8

Dependence of the absolute error in the total power carried by the optical beam on the refractive index contrast for the waveguide test structure using WA(6,6), θ = 0.5 rad , and d x = d z = 0.005 μ m .

Equations (16)

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2 Ψ z 2 + 2 Ψ x 2 + k 2 Ψ = 0 ,
( tan θ t + z ) 2 Ψ + ( 2 t 2 + k 2 ) Ψ = 0 .
( tan θ t + z ) Ψ = ± j 2 t 2 + k 2 Ψ .
Φ z = L [ 1 1 + ( L 2 M ) ] Φ
L = j β cos θ + tan θ t ,
M = k 2 β 2 + ( 1 + tan 2 θ ) 2 t 2 ,
Ψ = Φ exp ( j β cos θ z j β t sin θ ) .
1 + A = 1 + i = 1 n a i , n A 1 + b i , n A ,
Φ z = ( 1 2 L 1 M ) Φ ,
( L 2 1 4 M ) Φ z = ( 1 2 L M ) Φ ,
Φ z = i = 1 n a i , n L M L 2 + b i , n M Φ .
Φ ( z = z 0 + Δ z ) = exp ( i = 1 n a i , n L M L 2 + b i , n M Δ z ) Φ ( z = z 0 ) .
Φ ( z = z 0 + Δ z ) = i = 1 n exp ( a i , n L M L 2 + b i , n M Δ z ) Φ ( z = z 0 ) .
Φ ( z = z 0 + Δ z ) = i = 1 n L 2 + b i , n M Δ z 2 ( a i , n L M ) L 2 + b i , n M + Δ z 2 ( a i , n L M ) Φ ( z = z 0 ) .
L M = j β cos θ [ ( k 2 β 2 ) + ( 1 + tan 2 θ ) 2 t 2 ] + ( k 2 β 2 ) tan θ t + ( 1 + tan θ ) tan θ 3 t 3 ,
L 2 = β 2 cos 2 θ + 2 j β sin θ t + tan 2 θ 2 t 2 ,

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