Abstract

A new complex Jacobi iterative technique adapted for the solution of three-dimensional (3D) wide-angle (WA) beam propagation is presented. The beam propagation equation for analysis of optical propagation in waveguide structures is based on a novel modified Padé(1,1) approximant operator, which gives evanescent waves the desired damping. The resulting approach allows more accurate approximations to the true Helmholtz equation than the standard Padé approximant operators. Furthermore, a performance comparison of the traditional direct matrix inversion and this new iterative technique for WA-beam propagation method is reported. It is shown that complex Jacobi iteration is faster and better-suited for large problems or structures than direct matrix inversion.

© 2008 Optical Society of America

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Corrections

Khai Q. Le, R. Godoy-Rubio, Peter Bienstman, and G. Ronald Hadley, "The complex Jacobi iterative method for three-dimensional wide-angle beam propagation: erratum," Opt. Express 16, 21942-21942 (2008)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-26-21942

References

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  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]
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    [CrossRef] [PubMed]
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    [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. 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.E 82-C, 1154-1158 (1999).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
    [CrossRef]
  17. 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]
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    [CrossRef]
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    [CrossRef]
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2007 (2)

S. Sujecki, "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. 25, 138-145 (2007).
[CrossRef]

C. Ma and E. V. Keuren, "A three-dimensional wide-angle beam propagation method for optical waveguide structures," Opt. Express 15, 402-407 (2007).
[CrossRef] [PubMed]

2006 (2)

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

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

2005 (1)

G. R. Hadley, "A complex Jacobi iterative method for the indefinite Helmholtz equation," J. Comp. Phys. 203, 358-370 (2005).
[CrossRef]

2002 (1)

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)

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.E 82-C, 1154-1158 (1999).

1998 (2)

1997 (3)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997).
[CrossRef]

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]

Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

1994 (2)

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

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

1992 (2)

1990 (1)

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.E 82-C, 1154-1158 (1999).

Baets, R.

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

Benson, T. M.

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.E 82-C, 1154-1158 (1999).

Bérenger, J. P.

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

Bienstman, P.

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

Brooke, G. H.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997).
[CrossRef]

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]

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]

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.E 82-C, 1154-1158 (1999).

Ho, P. L.

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.E 82-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.E 82-C, 1154-1158 (1999).

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.

Koshiba, M.

Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

Lee, P. C.

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

Lu, Y. Y.

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, "A complex coefficient rational approximation of1+x ," Appl. Numer. Math. 27, 141-156 (1998).
[CrossRef]

Ma, C.

Maes, B.

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

Milinazzo, F. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997).
[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]

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]

Sewell, 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.E 82-C, 1154-1158 (1999).

Shiraishi, T.

Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

Sujecki, S.

S. Sujecki, "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. 25, 138-145 (2007).
[CrossRef]

Tsuji, Y.

Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

Vandersteengen, P.

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

Voges, E.

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

Zala, C. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997).
[CrossRef]

Appl. Numer. Math. (1)

Y. Y. Lu, "A complex coefficient rational approximation of1+x ," Appl. Numer. Math. 27, 141-156 (1998).
[CrossRef]

E (1)

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.E 82-C, 1154-1158 (1999).

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. Acoust. Soc. Am. (1)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, "Rational square root approximations for parabolic equation algorithms," J. Acoust. Soc. Am. 101, 760-766 (1997).
[CrossRef]

J. Comp. Phys. (1)

G. R. Hadley, "A complex Jacobi iterative method for the indefinite Helmholtz equation," J. Comp. Phys. 203, 358-370 (2005).
[CrossRef]

J. Comput. Phys. (1)

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

J. Lightwave Technol. (3)

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

Y. Tsuji, M. Koshiba, and T. Shiraishi, "Finite element beam propagation method for three-dimensional optical waveguide structures," J. Lightwave Technol. 15, 1728-1734 (1997).
[CrossRef]

G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

S. Sujecki, "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. 25, 138-145 (2007).
[CrossRef]

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

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. Express (2)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

P. Vandersteengen, B. Maes, P. Bienstman, and R. Baets, "Using the complex Jacobi method to simulate Kerr non-linear photonic components," Opt. Quantum Electron. 38, 35-44 (2006).
[CrossRef]

Other (2)

T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, 2006).
[CrossRef]

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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Figures (5)

Fig. 1.
Fig. 1.

The absolute values of (1+X)1/2-1 (solid line), its first-order standard Padé approximant (X/2)/(1+X/4) (solid line with circles) and modified Padé approximant (X/2)/(1+X/{4(1+ibeta/2)}) (dotted line).

Fig. 2.
Fig. 2.

The absolute value of (1+X)1/2-1 (solid line), the first-order standard (X/2)/(1+X/4) (solid line with circles) and modified (X/2)/(1+X/{4(1+ibeta/2)}) (dotted line) Padé approximant of (1+X)1/2-1 with respect to X.

Fig. 3.
Fig. 3.

The iteration count per propagation step for propagation through a symmetric Y-branch waveguide obtained using the CJI method for standard Padé approximant-based WA BPM without (1) and with (2) PML and the CJI method for the modified Padé without (3) and with (4) PML. (um is defined as µm)

Fig. 4.
Fig. 4.

Y-branch optical rib waveguide

Fig. 5.
Fig. 5.

Intensity contours for TE mode propagating in a 3D Y-branch rib waveguide at z=3µm calculated by (a) DMI and (b) the CJI method.

Tables (1)

Tables Icon

Table 1. Quantitative comparison of runtimes of the direct matrix inversion and the complex Jacobi iteration for WA beam propagation in waveguide (WG) structures

Equations (20)

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

H z i 2 k 2 H z 2 = i P 2 k H ,
H z = i P 2 k 1 i 2 k z H .
z n + 1 = i P 2 k 1 i 2 k z n .
H z i P 2 k 1 + P 4 k 2 H .
H z = i ( P + k 2 k ) H = i k ( 1 + X 1 ) H ,
1 + X 1 P 2 k 2 1 + P 4 k 2 = X 2 1 + X 4 .
i k z = P k 2 2 i k z .
f ( X ) = X 2 + f ( X ) ,
f n + 1 ( X ) = X 2 + f n ( X ) for n = 0,1,2 , . . .
f 0 ( X ) = i β where β > 0 .
H z i P 2 k 1 + P 4 k 2 ( 1 + i β 2 ) H ,
( 1 + ξ P ) ϕ n + 1 = ( 1 + ξ * P ) ϕ ,
( 2 + k 0 2 ( n 2 n ref 2 ) + 1 ξ ) ϕ n + 1 = ( ξ * ξ P + 1 ξ ) ϕ n ,
( 2 + k 0 2 ( n 2 n ref 2 ) + 4 k 0 2 n ref 2 1 + ( 1 + k 2 β Δ z / 2 ) β / 2 + i k Δ z ) ( 1 + k 2 β Δ z / 2 ) 2 + k 2 Δ z 2 ) ϕ n + 1 = source term .
A n + 1 ϕ i + 1 , j n + 1 + B n + 1 ϕ i 1 , j n + 1 + C n + 1 ϕ i , j n + 1 + D n + 1 ϕ i , j 1 n + 1 + E n + 1 ϕ i , j + 1 n + 1
= A n ϕ i + 1 , j n + B n ϕ i 1 , j n + C n ϕ i , j n + D n ϕ i , j 1 n + E n ϕ i , j + 1 n .
A n + 1 = B n + 1 = 1 Δ x 2 , D n + 1 = E n + 1 = 1 Δ y 2 ,
C n + 1 = k 0 2 ( n 2 n ref 2 ) + 1 ξ 2 ( 1 Δ x 2 + 1 Δ y 2 ) ,
A n = B n = ξ * ξ 1 Δ x 2 , D n = E n = ξ * ξ 1 Δ y 2 ,
C n = ξ * ξ ( k 0 2 ( n 2 n ref 2 ) + 1 ξ * 2 ( 1 Δ x 2 + 1 Δ y 2 ) .

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