Abstract

We present a new method for splitting of operators in the three-dimensional finite difference split-step nonparaxial beam propagation method. The method increases the accuracy and the efficiency in terms of speed and memory requirements for three-dimensional wide-angle beam propagation. It also makes the application of the perfectly matched layer (PML) boundary condition very simple.

© 2009 Optical Society of America

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  1. D. Yevick and M. Glasner, “Forward wide-angle light propagation in semiconductor, rib waveguides,” Opt. Lett. 15, 174-176 (1990).
    [CrossRef] [PubMed]
  2. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743-1745 (1992).
    [CrossRef] [PubMed]
  3. J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, “Efficient nonuniform scheme for paraxial and wide-angle finite difference beam propagation methods,” J. Lightwave Technol. 17, 677-683 (1999).
    [CrossRef]
  4. J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
    [CrossRef]
  5. I. Ilic, R. Scarmozzino, and R. Osgood, “Investigation of the Padé approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits,” J. Lightwave Technol. 14, 2813-2822 (1996).
    [CrossRef]
  6. Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p ?1)/p] Padé approximant of the propagator,” Opt. Lett. 27, 683-685 (2002).
    [CrossRef]
  7. Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photonics Technol. Lett. 14, 1533-1535 (2002).
    [CrossRef]
  8. P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photonics Technol. Lett. 13, 1316-1318 (2001).
    [CrossRef]
  9. A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082-1087 (2004).
    [CrossRef]
  10. A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 18, 944-946 (2006).
    [CrossRef]
  11. A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum Electron. 38, 19-34 (2006).
    [CrossRef]
  12. D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
    [CrossRef]
  13. S. L. Chui and Y. Y. Lu, “Wide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner,” J. Opt. Soc. Am. A 21, 420-425 (2004).
    [CrossRef]
  14. C. Ma and V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14, 4668-4674(2006).
    [CrossRef] [PubMed]
  15. C. Ma and V. Keuren, “A three-dimensional wide-angle BPM for optical waveguide structures,” Opt. Express 15, 402-407 (2007).
    [CrossRef] [PubMed]
  16. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
    [CrossRef]
  17. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
    [CrossRef]
  18. D. Bhattacharya and A. Sharma, “Split-step non-paraxial finite difference method for 3-D scalar wave propagation,” Opt. Quantum Electron. 39, 865-876 (2007).
    [CrossRef]
  19. A. Sharma, D. Bhattacharya, and A. Agrawal, “Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method,” Opt. Quantum Electron. 39, 623-626 (2007).
    [CrossRef]
  20. M. K. Jain, P. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation (Wiley Eastern, 1985).
  21. D. Yevick, “The application of complex Padé approximants to vector field propagation,” IEEE Photonics Technol. Lett. 12, 1636-1638 (2000).
    [CrossRef]
  22. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  23. C. Vasallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
    [CrossRef]
  24. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
    [CrossRef]
  25. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
    [CrossRef]
  26. D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
    [CrossRef]
  27. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599-604 (1994).
    [CrossRef]
  28. B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD--TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399-401 (1995).
    [CrossRef]
  29. A. Agrawal and A. Sharma, “Perfectly matched layer in numerical wave propagation: factors that affect its performance,” Appl. Opt. 43, 4225-4231 (2004).
    [CrossRef] [PubMed]
  30. D. Yevick, J. Yu, and F. Schmidt, “Analytical studies of absorbing and impedance-matched boundary layers,” IEEE Photonics Technol. Lett. 9, 73-75 (1997).
    [CrossRef]

2007 (3)

D. Bhattacharya and A. Sharma, “Split-step non-paraxial finite difference method for 3-D scalar wave propagation,” Opt. Quantum Electron. 39, 865-876 (2007).
[CrossRef]

A. Sharma, D. Bhattacharya, and A. Agrawal, “Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method,” Opt. Quantum Electron. 39, 623-626 (2007).
[CrossRef]

C. Ma and V. Keuren, “A three-dimensional wide-angle BPM for optical waveguide structures,” Opt. Express 15, 402-407 (2007).
[CrossRef] [PubMed]

2006 (5)

C. Ma and V. Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14, 4668-4674(2006).
[CrossRef] [PubMed]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 18, 944-946 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum Electron. 38, 19-34 (2006).
[CrossRef]

2004 (3)

2002 (2)

Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p ?1)/p] Padé approximant of the propagator,” Opt. Lett. 27, 683-685 (2002).
[CrossRef]

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photonics Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

2001 (2)

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photonics Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
[CrossRef]

2000 (1)

D. Yevick, “The application of complex Padé approximants to vector field propagation,” IEEE Photonics Technol. Lett. 12, 1636-1638 (2000).
[CrossRef]

1999 (1)

1997 (1)

D. Yevick, J. Yu, and F. Schmidt, “Analytical studies of absorbing and impedance-matched boundary layers,” IEEE Photonics Technol. Lett. 9, 73-75 (1997).
[CrossRef]

1996 (5)

C. Vasallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
[CrossRef]

J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
[CrossRef]

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

1995 (2)

D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
[CrossRef]

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD--TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399-401 (1995).
[CrossRef]

1994 (2)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1992 (1)

1990 (1)

1985 (1)

M. K. Jain, P. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation (Wiley Eastern, 1985).

Agrawal, A.

A. Sharma, D. Bhattacharya, and A. Agrawal, “Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method,” Opt. Quantum Electron. 39, 623-626 (2007).
[CrossRef]

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum Electron. 38, 19-34 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 18, 944-946 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082-1087 (2004).
[CrossRef]

A. Agrawal and A. Sharma, “Perfectly matched layer in numerical wave propagation: factors that affect its performance,” Appl. Opt. 43, 4225-4231 (2004).
[CrossRef] [PubMed]

Bardyszewski, W.

D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

Bhattacharya, D.

D. Bhattacharya and A. Sharma, “Split-step non-paraxial finite difference method for 3-D scalar wave propagation,” Opt. Quantum Electron. 39, 865-876 (2007).
[CrossRef]

A. Sharma, D. Bhattacharya, and A. Agrawal, “Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method,” Opt. Quantum Electron. 39, 623-626 (2007).
[CrossRef]

Chen, B.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
[CrossRef]

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD--TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399-401 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Chui, S. L.

Collino, F.

C. Vasallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

Fang, D. G.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
[CrossRef]

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD--TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399-401 (1995).
[CrossRef]

Glasner, M.

D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
[CrossRef]

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

Hadley, G. R.

Ho, P. L.

Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p ?1)/p] Padé approximant of the propagator,” Opt. Lett. 27, 683-685 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photonics Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

Huang, W. P.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
[CrossRef]

Ilic, I.

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

Iyengar, P. K.

M. K. Jain, P. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation (Wiley Eastern, 1985).

Jain, M. K.

M. K. Jain, P. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation (Wiley Eastern, 1985).

Jain, R. K.

M. K. Jain, P. K. Iyengar, and R. K. Jain, Numerical Methods for Scientific and Engineering Computation (Wiley Eastern, 1985).

Keuren, V.

Lu, Y. Y.

S. L. Chui and Y. Y. Lu, “Wide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner,” J. Opt. Soc. Am. A 21, 420-425 (2004).
[CrossRef]

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photonics Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p ?1)/p] Padé approximant of the propagator,” Opt. Lett. 27, 683-685 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photonics Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
[CrossRef]

Ma, C.

Matsubara, K.

Nakano, H.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

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

J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
[CrossRef]

Osgood, R.

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

Scarmozzino, R.

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

Schmidt, F.

D. Yevick, J. Yu, and F. Schmidt, “Analytical studies of absorbing and impedance-matched boundary layers,” IEEE Photonics Technol. Lett. 9, 73-75 (1997).
[CrossRef]

Sekiguchi, M.

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

J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
[CrossRef]

Sharma, A.

D. Bhattacharya and A. Sharma, “Split-step non-paraxial finite difference method for 3-D scalar wave propagation,” Opt. Quantum Electron. 39, 865-876 (2007).
[CrossRef]

A. Sharma, D. Bhattacharya, and A. Agrawal, “Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method,” Opt. Quantum Electron. 39, 623-626 (2007).
[CrossRef]

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum Electron. 38, 19-34 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 18, 944-946 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082-1087 (2004).
[CrossRef]

A. Agrawal and A. Sharma, “Perfectly matched layer in numerical wave propagation: factors that affect its performance,” Appl. Opt. 43, 4225-4231 (2004).
[CrossRef] [PubMed]

Shibayama, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
[CrossRef]

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

J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
[CrossRef]

Takahashi, T.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
[CrossRef]

Vasallo, C.

C. Vasallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Wei, S. H.

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photonics Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Xu, C. L.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang, and B. Chen, “The perfectly matched layer boundary condition for scalar finite-difference time-domain method,” IEEE Photonics Technol. Lett. 13, 454-456 (2001).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
[CrossRef]

Yamauchi, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photonics Technol. Lett. 18, 661-663 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” IEEE Photonics Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

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

J. Yamauchi, J. Shibayama, M. Sekiguchi, and H. Nakano, “Improved multistep method for wide-angle beam propagation,” IEEE Photonics Technol. Lett. 8, 1361-1363 (1996).
[CrossRef]

Yevick, D.

D. Yevick, “The application of complex Padé approximants to vector field propagation,” IEEE Photonics Technol. Lett. 12, 1636-1638 (2000).
[CrossRef]

D. Yevick, J. Yu, and F. Schmidt, “Analytical studies of absorbing and impedance-matched boundary layers,” IEEE Photonics Technol. Lett. 9, 73-75 (1997).
[CrossRef]

D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
[CrossRef]

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

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photonics Technol. Lett. 8, 649-651 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photonics Technol. Lett. 8, 652-654 (1996).
[CrossRef]

Yu, J.

D. Yevick, J. Yu, and F. Schmidt, “Analytical studies of absorbing and impedance-matched boundary layers,” IEEE Photonics Technol. Lett. 9, 73-75 (1997).
[CrossRef]

Yu, Jun

D. Yevick, Jun Yu, W. Bardyszewski, and M. Glasner, “Stability issues in vector electric field propagation,” IEEE Photonics Technol. Lett. 7, 658-660 (1995).
[CrossRef]

Zhou, B. H.

B. Chen, D. G. Fang, and B. H. Zhou, “Modified Berenger PML absorbing boundary condition for FD--TD meshes,” IEEE Microwave Guided Wave Lett. 5, 399-401 (1995).
[CrossRef]

Zhou, D.

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Figures (7)

Fig. 1
Fig. 1

Comparison of error with propagation distance for three different splitting methods for propagation of the fundamental mode in a straight θ = 0 ° square core ( 5 μm × 5 μm ) waveguide.

Fig. 2
Fig. 2

Comparison of error with propagation distance for three different splitting methods for propagation of the fundamental mode in a square core ( 5 μm × 5 μm ) waveguide tilted at 30 ° to the z axis on the x–z plane.

Fig. 3
Fig. 3

Comparison between the intensity profiles of the exact field and the fields computed using the various methods after 40 μm propagation.

Fig. 4
Fig. 4

Error versus n r plot for 40 μm propagation of the fundamental mode of a square core waveguide tilted at 30 ° to the z axis on the x–z plane.

Fig. 5
Fig. 5

Absorption of a Gaussian beam, propagating in a uniform medium at θ = 15 ° to the z axis, in a PML applied along the x direction. In a numerical window of 150 points, the PML is 22 points. We have used λ = 1 μm , Δ x = 0.7 μm and p 0 = 200 .

Fig. 6
Fig. 6

Variation of reflectivity with angle θ made with the z axis. The PML is optimized for absorption at θ = 15 ° . PML   width = 22   points , p 0 = 200 , numerical window = 150   points , λ = 1 μm , and Δ x = 0.7 μm .

Fig. 7
Fig. 7

Variation of reflectivity with profile factor p 0 for different propagation angles. The thin curves represent results obtained using the analytical method proposed by Yevick et al. [30]. The thick curves represent results obtained using our method.

Equations (24)

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2 ψ x 2 + 2 ψ y 2 + 2 ψ z 2 + k 0 2 n 2 ( x , y , z ) ψ ( x , y , z ) = 0 ,
Φ ( z ) z = H Φ ( z ) ,
Φ ( z ) = [ ψ ψ z ] ,
H = [ 0 1 ( 2 x 2 + 2 y 2 + k 0 2 n 2 ) 0 ] ,
H ( z ) = H ^ 1 + H ^ 2 ( z ) = [ 0 1 ( 2 x 2 + 2 y 2 + k 0 2 n r 2 ) 0 ] + [ 0 0 k 0 2 ( n r 2 n 2 ) 0 ] .
Φ ( z + Δ z ) = P Q ( z ) P Φ ( z ) + O [ ( Δ z ) 3 ] ,
H ( z ) = H 1 + H 2 ( z ) = [ 0 1 ( 2 y 2 + k 0 2 n r 2 ) 0 ] + [ 0 0 2 x 2 + k 0 2 ( n r 2 n 2 ( z ) ) 0 ] ,
2 y 2 = 1 Δ y 2 [ 2 sinh 1 ( δ y 2 ) ] 2 = 1 Δ y 2 [ δ y 2 1 12 δ y 4 + 1 90 δ y 6 ] ,
P = ( V y 0 0 V y ) ( cos ( Ω Δ z / 2 ) Ω - 1 sin ( Ω Δ z / 2 ) Ω sin ( Ω Δ z / 2 ) cos ( Ω Δ z / 2 ) ) ( V y 0 0 V y ) ,
Q P Φ = [ φ 1 φ 1 A Δ z R Δ z + φ 2 ] ,
Q P Φ = [ φ 1 φ 1 A Δ z B φ 1 Δ z R Δ z + φ 2 ] .
CF = | ψ exact * ψ calc d x | 2 | ψ inp | 2 d x · | ψ exact | 2 d x ,
x = h ( σ ) , y = g ( η ) ,
h ( σ ) = σ , σ x p = x p + ξ x [ 1 i p x ( ξ x ) ] d ξ x , x p < σ x b ,
g ( η ) = η , η y p = y p + ξ y [ 1 i p y ( ξ y ) ] d ξ y , y p η y b .
ψ = h ( σ ) g ( η ) U ( σ , η , z ) .
2 U z 2 + F σ 2 U σ 2 + F η 2 U η 2 + ( G σ + G η + k 0 2 n 2 ( σ , η , z ) ) U = 0 ,
F σ = [ h ( σ ) ] 2 , F η = [ g ( η ) ] 2 , G σ = 1 2 h 4 [ h h - 3 2 h 2 ] , G η = 1 2 g 4 [ g g - 3 2 g 2 ] ,
Q P Φ = [ φ 1 φ 1 A F σ Δ z F η B φ 1 Δ z R Δ z G η φ 1 Δ z φ 1 G σ Δ z + φ 2 ] ,
Q = [ I 0 R ˜ I ] ,
P Φ = [ Φ 1 Φ 2 ] ,
QP Φ = [ Φ 1 R ˜ Φ 1 + Φ 2 ] ,
P Φ = [ φ 1 φ 2 ] ,
P Φ = [ φ 1 φ 2 ] ,

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