Abstract

The development of three dimensional (3-D) waveguide structures for chip scale planar lightwave circuits (PLCs) is hampered by the lack of effective 3-D wide-angle (WA) beam propagation methods (BPMs). We present a simple 3-D wide-angle beam propagation method (WA-BPM) using Hoekstra’s scheme along with a new 3-D wave equation splitting method. The applicability, accuracy and effectiveness of our method are demonstrated by applying it to simulations of wide-angle beam propagation and comparing them with analytical solutions.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  5. 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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  9. C. Vassallo, "Reformulation for the beam-propagation method," J. Opt. Soc. Am. A 10, 2208-2216 (1993).
    [CrossRef]
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    [CrossRef]
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    [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).
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  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).
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    [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. J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
    [CrossRef]
  21. S. Sujecki, P. Sewell, T. M. Benson, and P. C. Kendall, "Novel beam propagation algorithms for tapered optical structures," J. Lightwave Technol. 17, 2379-2388 (1999).
    [CrossRef]
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  23. D. Z. Djurdjevic, T. M. Benson,  et al, "Fast and accurate analysis of 3-D curved optical waveguide couplers," J. Lightwave Technol. 22, 2333-2340 (2004).
    [CrossRef]

2004 (3)

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]

1999 (1)

1998 (1)

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]

1996 (1)

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[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]

1992 (4)

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]

1991 (3)

1990 (2)

1981 (1)

1978 (1)

Agrawal, A.

Benson, T. M.

Chang, H. C.

Chiou, Y. P.

Chui, S. L.

Chung, Y.

Dagli, N.

Djurdjevic, D. Z.

Feit, M. D.

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]

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]

Kendall, P. C.

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.

Nakano, H.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[CrossRef]

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.

Saito, O.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[CrossRef]

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]

Sewell, P.

Sharma, A.

Shibayama, J.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[CrossRef]

Sujecki, S.

Uchiyama, O.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[CrossRef]

van der Donk, J.

Van Roey, J.

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]

Yamauchi, J.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[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]

J. Lightwave Technol. (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]

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

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol. 14, 2401-2406 (1996).
[CrossRef]

S. Sujecki, P. Sewell, T. M. Benson, and P. C. Kendall, "Novel beam propagation algorithms for tapered optical structures," J. Lightwave Technol. 17, 2379-2388 (1999).
[CrossRef]

D. Z. Djurdjevic, T. M. Benson,  et al, "Fast and accurate analysis of 3-D curved optical waveguide couplers," J. Lightwave Technol. 22, 2333-2340 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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]

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. Lett. (6)

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 (2)

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).

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

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Figures (3)

Fig. 1.
Fig. 1.

Modulus profiles of the input and outputs resulting from the propagation of a 3-D Gaussian beam with a 30-degree phase tilt a distance of 60 µm through free space. The beam is initially centered at (x, y, z)=(0, 0, 0), as shown in yellow with pink meshes. The paraxial output is in red, the wide-angle output is in green and the exact (analytical) output is in blue.

Fig. 2.
Fig. 2.

Modulus profiles of the input and outputs resulting from the propagation of the mode in a single-mode waveguide channel with a 30-degree tilt with respect to z axis a distance of 60 µm. The beam is initially centered at (x, y, z)=(0, 0, 0), as shown in yellow with pink meshes. The paraxial output is in red, the wide-angle output is in green and the exact (analytical) output is in blue.

Fig. 3.
Fig. 3.

(a) The relative L2 norm error; (b) the relative position shifts. All are calculated with respect to the exact solutions of the field modulus at z=60 µm using different Δx (Δy is set to equal Δx). The calculations using our wide-angle BPM are in green, and calculations using the classical BPM are in red. The plots with triangles are for the Gaussian beam propagation, and those with squares are for the beam propagation in waveguide.

Equations (19)

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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 z 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 Δ z ( ψ z ) l + 1 ( ψ z ) l Δ z = 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
ia ψ l + 1 ψ l + 1 2 + ψ l + 1 2 ψ l Δ z ( ψ z ) l + 1 ( ψ z ) l + 1 2 + ( ψ z ) l + 1 2 ( ψ z ) l Δ z
= 2 x 2 ( 2 ψ l + 1 2 2 ) + 2 y 2 ( ψ l + 1 + ψ l 2 ) + ( b ψ ) l + 1 + 2 ( b ψ ) l + 1 2 + ( b ψ ) l 4
ia ψ l + 1 2 ψ l Δ z ( ψ z ) l + 1 2 ( ψ z ) l Δ z = 1 2 2 ψ l + 1 2 x 2 + 1 2 2 ψ l y 2 + ( b ψ ) l + 1 2 + ( b ψ ) l 4
ia ψ l + 1 ψ l + 1 2 Δ z ( ψ z ) l + 1 ( ψ z ) l + 1 2 Δ z = 1 2 2 ψ l + 1 2 x 2 + 1 2 2 ψ l + 1 y 2 + ( b ψ ) l + 1 + ( b ψ ) l + 1 2 4
ia z ( ψ l + 1 + ψ l 2 ) = 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
ia z ( ψ l + 1 + 2 ψ l + 1 2 + ψ l 4 ) = 2 x 2 ( 2 ψ l + 1 2 2 )
+ 2 y 2 ( ψ l + 1 + ψ l 2 ) + ( b ψ ) l + 1 + 2 ( b ψ ) l + 1 2 + ( b ψ ) l 4
ia z ( ψ l + 1 2 + ψ l 4 ) = 1 2 2 ψ l + 1 2 x 2 + 1 2 2 ψ l y 2 + ( b ψ ) l + 1 2 + ( b ψ ) l 4
ia z ( ψ l + 1 + ψ l + 2 4 ) = 1 2 2 ψ l + 1 2 x 2 + 1 2 2 ψ l + 1 y 2 + ( b ψ ) l + 1 + ( b ψ ) l + 1 2 4
ia ψ l z = 2 2 ψ l y 2 + ( b ψ ) l
ia ψ l + 1 2 z = 2 2 ψ l + 1 2 x 2 + ( b ψ ) l + 1 2
ia ψ l + 1 z = 2 2 ψ l + 1 y 2 + ( b ψ ) l + 1
P x + ψ m 1 , j l + 1 2 + Q x , m , j + ψ m , j l + 1 2 + P x + ψ m + 1 , j l + 1 2 = P y ψ m , j 1 l + Q y , m , j ψ m , j l + P y ψ m , j + 1 l
P y + ψ m , j 1 l + 1 + Q y , m , j + ψ m , j l + 1 + P y + ψ m , j + 1 l + 1 = P x ψ m 1 , j l + 1 2 + Q x , m , j ψ m , j l + 1 2 + P x ψ m + 1 , j l + 1 2

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