Abstract

A new wide-angle (WA) beam propagation method (BPM) is developed whereby the exact scalar Helmholtz propagator is replaced by any one of a sequence of higher-order (m,n) Padé approximant operators. Unlike the previous well-known WA-BPM proposed by Hadley [Opt. Lett. 17, 1426 (1992)] , the resulting formulations allow one a direct solution of the second-order scalar wave equation without having to make slowly varying envelope approximations so that the WA formulations are completely general. The accuracy and improvement of this approximate calculation of the propagator is demonstrated in comparison with the exact result and existing approximate approaches. The method is employed to simulate two-dimensional (2D) and three-dimensional (3D) optical waveguides and compared with the results obtained by the existing approach.

© 2009 Optical Society of America

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References

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  1. 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]
  2. 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).
  3. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426-1428 (1992).
    [CrossRef] [PubMed]
  4. K. Q. Le, R. G. Rubio, P. Bienstman, and G. R. Hadley, “The complex Jacobi iterative method for three-dimensional wide-angle beam propagation,” Opt. Express 16, 17021-17030 (2008).
    [CrossRef] [PubMed]
  5. S. Sujecki, “Wide-angle, finite-difference beam propagation in oblique coordinate system,” J. Opt. Soc. Am. A 25, 138-145 (2007).
    [CrossRef]
  6. P. C. Lee and E. Voges, “Three-dimensional semi-vectorial wide-angle beam propagation method,” J. Lightwave Technol. 12, 215-225 (1994).
    [CrossRef]
  7. 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]
  8. A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. A 21, 1082-1087 (2004).
    [CrossRef]
  9. A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. Technol. Lett. 18, 944-946 (2006).
    [CrossRef]
  10. G. A. Baker, Essentials of Padé Approximants (Academic, 1975).
  11. J. H. Mathews, “Module for Padé approximation,” http://math.fullerton.edu/mathews/n2003/pade/PadeApproximationMod/Links/PadeApproximationMod_lnk_4.html.
  12. 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]
  13. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743-1745 (1992).
    [CrossRef] [PubMed]
  14. 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]

2008 (1)

2007 (2)

2006 (1)

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

2004 (1)

2002 (1)

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. E82-C, 1154-1158 (1999).

1997 (2)

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]

1994 (1)

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

1992 (2)

Agrawal, A.

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. 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]

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).

Baker, G. A.

G. A. Baker, Essentials of Padé Approximants (Academic, 1975).

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. E82-C, 1154-1158 (1999).

Bienstman, P.

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]

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]

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]

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).

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. 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).

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.

Le, K. Q.

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.

Ma, C.

Mathews, J. H.

J. H. Mathews, “Module for Padé approximation,” http://math.fullerton.edu/mathews/n2003/pade/PadeApproximationMod/Links/PadeApproximationMod_lnk_4.html.

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]

Rubio, R. G.

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. E82-C, 1154-1158 (1999).

Sharma, A.

A. Sharma and A. Agrawal, “A new finite-difference-based method for wide-angle beam propagation,” IEEE Photon. 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]

Sujecki, S.

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]

IEEE Photon. Technol. Lett. (1)

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

IEICE Trans. Electron. (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. E82-C, 1154-1158 (1999).

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. Lightwave Technol. (1)

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

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

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)

Other (2)

G. A. Baker, Essentials of Padé Approximants (Academic, 1975).

J. H. Mathews, “Module for Padé approximation,” http://math.fullerton.edu/mathews/n2003/pade/PadeApproximationMod/Links/PadeApproximationMod_lnk_4.html.

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Figures (5)

Fig. 1
Fig. 1

Absolute values of X (solid curve), the most useful low-order traditional Padé approximants, Hadley ( m , n ) , and K P ( m , n ) of X (dotted curves).

Fig. 2
Fig. 2

Absolute values of X (solid curve), the first-order (solid curve with circles), and rotated (dotted curve) K P ( 1 , 1 ) approximant of X.

Fig. 3
Fig. 3

Medium loss of WA beam propagation using CJI based on Hadley(1,1) (dotted curve) and K P ( 1 , 1 ) approximant (solid curve).

Fig. 4
Fig. 4

Input beam at z = 0 (solid curve that peaks in center) and output beam at z = 21 μ m in a 2D Y-branch rib waveguide calculated by WA-BPM based on K P ( 1 , 1 ) (solid curve that peaks on left and right) and Hadley(1,1) (circles).

Fig. 5
Fig. 5

Magnitude of TE fundamental mode after propagating 3 μ m in a 3D Y-branch rib waveguide calculated by WA-BPM based on (a) K P ( 1 , 1 ) and (b) Hadley(1,1).

Tables (1)

Tables Icon

Table 1 Most Useful Low-Order Padé Approximants for Helmholtz Propagator in Terms of the Operator X = P k 2

Equations (11)

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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 = 0 ,
2 Ψ z 2 = P Ψ ,
i 2 k 2 Ψ z 2 + Ψ z = i P 2 k Ψ + Ψ z ,
Ψ z = i P 2 k + z 1 i 2 k z Ψ .
z n + 1 = i P 2 k + z n 1 i 2 k z n .
Ψ z i k N ( m ) D ( n ) Ψ ,
Ψ z = i P Ψ = i k X Ψ ,
X N ( m ) D ( n ) .
D ( Ψ n + 1 Ψ n ) = i k Δ z 2 N ( Ψ n + 1 + Ψ n ) .
( 1 + ξ Hadley P Hadley ) Ψ n + 1 = ( 1 + ξ Hadley * P Hadley ) Ψ n ,
( 1 + ξ K P P K P ) Ψ n + 1 = ( 1 + ξ K P * P K P ) Ψ n ,

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