Abstract

Algorithms for effective modeling of optical propagation in three- dimensional waveguide structures are critical for the design of photonic devices. We present a three-dimensional (3-D) wide-angle beam propagation method (WA-BPM) using Hoekstra’s scheme. A sparse matrix algebraic equation is formed and solved using iterative methods. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation, along with a technique for shifting the simulation window to reduce the dimension of the numerical equation and a threshold technique to further ensure its convergence. These techniques can ensure the implementation of iterative methods for waveguide structures by relaxing the convergence problem, which will further enable us to develop higher-order 3-D WA-BPMs based on Padé approximant operators.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  2. D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994).
    [CrossRef]
  3. J. Van Roey, J. van der Donk, and P. E. Lagasse, "Beam-propagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803-810 (1981).
    [CrossRef]
  4. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
    [CrossRef]
  5. Y. Chung and N. Dagli, "An assessment of finite difference beam propagation," J. Quantum. Electron. 26, 1335-1339 (1990).
    [CrossRef]
  6. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).
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    [CrossRef] [PubMed]
  8. M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990).
    [CrossRef]
  9. C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993).
    [CrossRef]
  10. J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991).
    [CrossRef]
  11. R. P. Ratowsky and J. A. FleckJr., "Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction," Opt. Lett. 16, 787-789 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994).
    [CrossRef]
  14. A. Sharma and A. Agrawal, "New method for nonparaxial beam propagation," J. Opt. Soc. Am. A 21, 1082-1087 (2004).
    [CrossRef]
  15. 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]
  16. G. R. Hadley, "Wide-angle beam propagation using Padé approximant operators," Opt. Lett. 17, 1426-1428 (1992).
    [CrossRef] [PubMed]
  17. G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992).
    [CrossRef] [PubMed]
  18. 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]
  19. Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
    [CrossRef]
  20. C. Ma and E. V. Keuren, "A simple three dimensional wide-angle beam propagation method," Opt. Express 14, 4668-4674 (2006).
    [CrossRef] [PubMed]
  21. W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992).
    [CrossRef]

2006 (1)

2004 (2)

2000 (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

1997 (1)

Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
[CrossRef]

1994 (2)

P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994).
[CrossRef]

D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994).
[CrossRef]

1993 (1)

C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993).
[CrossRef]

1992 (5)

P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992).
[CrossRef]

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

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

W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992).
[CrossRef]

1991 (3)

1990 (2)

M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990).
[CrossRef]

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

1981 (1)

1978 (1)

Agrawal, A.

Chui, S. L.

Chung, Y.

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

Dagli, N.

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

Feit, M. D.

M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990).
[CrossRef]

M. D. Feit and J. A. FleckJr., "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978).
[CrossRef] [PubMed]

Feng, E.

Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
[CrossRef]

Fleck, J. A.

Fu, J.

Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
[CrossRef]

Gerdes, J.

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

Hadley, G. R.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

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]

Huang, W. P.

W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992).
[CrossRef]

Ju, Z.

Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
[CrossRef]

Keuren, E. V.

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]

Lagasse, P. E.

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]

Lee, P. -C.

P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994).
[CrossRef]

P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992).
[CrossRef]

Lu, Y. Y.

Ma, C.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991).
[CrossRef]

Ratowsky, R. P.

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

Schulz, D.

P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992).
[CrossRef]

Sharma, A.

van der Donk, J.

Van Roey, J.

Vassallo, C.

C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993).
[CrossRef]

Voges, E.

P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994).
[CrossRef]

P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992).
[CrossRef]

Xu, C. L.

W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992).
[CrossRef]

Yevick, D.

D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994).
[CrossRef]

Appl. Opt. (1)

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

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-161 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method," IEEE Photon. Technol. Lett. 4, 1118-1120 (1992).
[CrossRef]

J. Lightwave Technol. (2)

P. -C. Lee, D. Schulz, and E. Voges, "Three-dimensional finite difference beam propagation algorithms for photonic devices," J. Lightwave Technol. 10, 1832-1838 (1992).
[CrossRef]

P. -C. Lee and E. Voges, "Three-dimensional semi-vectorial wave-angle beam propagation method," J. Lightwave Technol. 12, 215-225 (1994).
[CrossRef]

J. Opt. Soc. Am A (1)

M. D. Feit and J. A. FleckJr., "Analysis of rib waveguides and couplers by the propagation method," J. Opt. Soc. Am A 7, 73-79 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Amer. A (1)

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Amer. A 8, 389-394 (1991).
[CrossRef]

J. Quantum. Electron. (1)

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

J.Opt. Soc. Am. A (1)

C. Vassallo, "Reformulation for the beam-propagation method," J.Opt. Soc. Am. A 10, 2208-2216 (1993).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

Z. Ju, J. Fu and E. Feng, "A simple wide-angle beam-propagation method for integrated optics," Microwave Opt. Technol. Lett. 14, 345-347 (1997).
[CrossRef]

Opt. Commun. (1)

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]

Opt. Express (1)

Opt. Lett. (4)

Opt. Quantum. Electron. (1)

D. Yevick, "A guide to electric field propagation techniques for guided-wave optics," Opt. Quantum. Electron. 26, S185-S197 (1994).
[CrossRef]

Other (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical recipes: The art of scientific computing, (Cambridge University Press, New York, 1986).

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Tables (1)

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Table 1. The relative L2 norm errors and relative position shift

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

2 E x 2 + 2 E y 2 + 2 E z 2 + k 0 2 n 2 x y z E = 0
E x y z = ψ x y z exp ( ik 0 n 0 z )
ia ψ 2 ψ z 2 = 2 ψ x 2 + 2 ψ y 2 + b ψ
ia ψ z = 2 ψ x 2 + 2 ψ y 2 + b ψ
ia ψ l + 1 ψ l Δz 1 Δz [ ψ z l + 1 ψ z l ]
= 2 x 2 ( ψ l + 1 + ψ l 2 ) + 2 y 2 ( ψ l + 1 + ψ l 2 ) + ( b ψ ) l + 1 + ( b ψ ) l 2
ψ z l = 1 ia [ 2 ψ l x 2 + 2 ψ l y 2 + ( b ψ ) l ]
ψ z l + 1 = 1 ia [ 2 ψ l + 1 x 2 + 2 ψ l + 1 y 2 + ( b ψ ) l + 1 ]
A ψ m 1 , j l + 1 + B ψ m , j 1 l + 1 + C ψ m , j l + 1 + D ψ m , j + 1 l + 1 + E ψ m + 1 , j l + 1
= R ψ m 1 , j l + S ψ m , j 1 l + T ψ m , j l + U ψ m , j + 1 l + V ψ m + 1 + j l
{ A = E = 1 Δ x 2 1 + 2 i a Δ z B = D = 1 Δ y 2 1 + 2 i a Δ z C = 1 Δ x 2 + 1 Δ y 2 2 4 i a Δ z + 2 ia Δ z 1 2 i a Δ z b m , j l + 1 R = V = 1 Δ x 2 + 1 + 2 i a Δ z S = U = 1 Δ y 2 + 1 + 2 i a Δ z T = 1 Δ x 2 + 1 Δ y 2 2 4 i a Δ z + 2 ia Δ z + 1 + 2 i a Δ z b m , j i

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