Abstract

Wide-angle full-vector beam propagation methods (BPMs) for three-dimensional wave-guiding structures can be derived on the basis of rational approximants of a square root operator or its exponential (i.e., the one-way propagator). While the less accurate BPM based on the slowly varying envelope approximation can be efficiently solved by the alternating direction implicit (ADI) method, the wide-angle variants involve linear systems that are more difficult to handle. We present an efficient solver for these linear systems that is based on a Krylov subspace method with an ADI preconditioner. The resulting wide-angle full-vector BPM is used to simulate the propagation of wave fields in a Y branch and a taper.

© 2004 Optical Society of America

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    [CrossRef] [PubMed]
  2. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef]
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    [CrossRef]
  4. H. J. W. M. Hoekstra, “On beam propagation methods for modelling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
    [CrossRef]
  5. R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
    [CrossRef]
  6. D. Yevick, J. Yu, W. Bardyszewski, M. Glasner, “Stability issues in vector electric-field propagation,” IEEE Photon. Technol. Lett. 7, 658–660 (1995).
    [CrossRef]
  7. D. Yevick, “The application of complex Padé approximants to vector field propagation,” IEEE Photon. Technol. Lett. 12, 1636–1638 (2000).
    [CrossRef]
  8. W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
    [CrossRef]
  9. I. Mansour, A. D. Capobianco, C. Rosa, “Noniterative vectorial beam propagation method with a smoothing digital filter,” J. Lightwave Technol. 14, 908–913 (1996).
    [CrossRef]
  10. E. E. Kriezis, A. G. Papagiannakis, “A three-dimensional full vectorial beam propagation method for z-dependent structures,” IEEE J. Quantum Electron. 33, 883–890 (1997).
    [CrossRef]
  11. E. Montanari, S. Selleri, L. Vincetti, M. Zoboli, “Finite-element full-vectorial propagation analysis for three-dimensional z-varying optical waveguides,” J. Lightwave Technol. 16, 703–714 (1998).
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    [CrossRef]
  14. R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
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  18. H. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631–644 (1992).
    [CrossRef]

2002 (1)

2001 (1)

2000 (2)

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

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

1999 (1)

1998 (2)

1997 (2)

E. E. Kriezis, A. G. Papagiannakis, “A three-dimensional full vectorial beam propagation method for z-dependent structures,” IEEE J. Quantum Electron. 33, 883–890 (1997).
[CrossRef]

H. J. W. M. Hoekstra, “On beam propagation methods for modelling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
[CrossRef]

1996 (1)

I. Mansour, A. D. Capobianco, C. Rosa, “Noniterative vectorial beam propagation method with a smoothing digital filter,” J. Lightwave Technol. 14, 908–913 (1996).
[CrossRef]

1995 (1)

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

1994 (1)

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

1993 (1)

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

1992 (2)

H. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631–644 (1992).
[CrossRef]

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

1991 (1)

1978 (1)

Bardyszewski, W.

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

Barrett, R.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Berry, M. W.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Capobianco, A. D.

I. Mansour, A. D. Capobianco, C. Rosa, “Noniterative vectorial beam propagation method with a smoothing digital filter,” J. Lightwave Technol. 14, 908–913 (1996).
[CrossRef]

Chan, T. F.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Chang, H.-C.

Demmel, J.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Donato, J.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Dongarra, J.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Eijkhout, V.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Feit, M. D.

Fleck, J. A.

Glasner, M.

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

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Hadley, G. R.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Ho, P. L.

Hoekstra, H. J. W. M.

H. J. W. M. Hoekstra, “On beam propagation methods for modelling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
[CrossRef]

Hsueh, Y.-L.

Huang, W. P.

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Kriezis, E. E.

E. E. Kriezis, A. G. Papagiannakis, “A three-dimensional full vectorial beam propagation method for z-dependent structures,” IEEE J. Quantum Electron. 33, 883–890 (1997).
[CrossRef]

Law, C. T.

Lu, Y. Y.

Luo, Q.

Mansour, I.

I. Mansour, A. D. Capobianco, C. Rosa, “Noniterative vectorial beam propagation method with a smoothing digital filter,” J. Lightwave Technol. 14, 908–913 (1996).
[CrossRef]

Montanari, E.

Nakano, H.

Papagiannakis, A. G.

E. E. Kriezis, A. G. Papagiannakis, “A three-dimensional full vectorial beam propagation method for z-dependent structures,” IEEE J. Quantum Electron. 33, 883–890 (1997).
[CrossRef]

Pozo, R.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Ratowsky, R. P.

Romine, C.

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Rosa, C.

I. Mansour, A. D. Capobianco, C. Rosa, “Noniterative vectorial beam propagation method with a smoothing digital filter,” J. Lightwave Technol. 14, 908–913 (1996).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Selleri, S.

Takahashi, G.

Van der Vorst, H.

H. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631–644 (1992).
[CrossRef]

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

Vincetti, L.

Xu, C. L.

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Yamauchi, J.

Yang, M.-C.

Yevick, D.

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

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

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

Yu, J.

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

Zoboli, M.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

W. P. Huang, C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam-propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

E. E. Kriezis, A. G. Papagiannakis, “A three-dimensional full vectorial beam propagation method for z-dependent structures,” IEEE J. Quantum Electron. 33, 883–890 (1997).
[CrossRef]

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

R. Scarmozzino, A. Gopinath, R. Pregla, S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

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

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

J. Lightwave Technol. (4)

Opt. Lett. (4)

Opt. Quantum Electron. (2)

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

H. J. W. M. Hoekstra, “On beam propagation methods for modelling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
[CrossRef]

SIAM J. Sci. Statist. Comput. (1)

H. Van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631–644 (1992).
[CrossRef]

Other (1)

R. Barrett, M. W. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).

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Figures (6)

Fig. 1
Fig. 1

Example 1: a Y branch.

Fig. 2
Fig. 2

Magnitude of Hy at z=L for the Y branch based on the [2/3] Padé approximant of the propagator.

Fig. 3
Fig. 3

Magnitude of Hy at z=L for the Y branch calculated by the θ method.

Fig. 4
Fig. 4

Example 2: a linear taper.

Fig. 5
Fig. 5

Magnitude of Hy at z=L for the taper based on the [3/4] Padé approximant of the propagator.

Fig. 6
Fig. 6

Magnitude of Hy at z=L for the taper calculated by the θ method.

Equations (29)

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

(1+bX)w=f
2uz2+Cu0,
C11=n2x1n2x+2y2+k02n2,C12=n2x1n2y-2xy,C21=n2y1n2x-2yx,C22=2x2+n2y1n2y+k02n2,
uz=ik0n*1+Xu.
uj+1=Puj,    P=exp(is 1+X),
Pk=1pak1+bkX.
uj+1=k=1pak(1+bkX)-1uj.
(1+bkX)wk=uj.
zϕ=ik0n*2Xϕ.
ϕj+1=1+[(is)/4]X1-[(is)/4]Xϕj,
X=X1+X2=X11(1)0X21X22(1)+X11(2)X120X22(2),
X11(1)=1k02n*2n2x1n2x+12(k02n2-k02n*2),X11(2)=1k02n*22y2+12(k02n2-k02n*2),X12=1k02n*2n2x1n2y-2xy,X21=1k02n*2n2y1n2x-2yx,X22(1)=1k02n*22x2+12(k02n2-k02n*2),X22(2)=1k02n*2n2y1n2y+12(k02n2-k02n*2).
ϕj+1=1-is4X2-11+[(is)/4]X11-[(is)/4]X11+is4X2ϕj.
1-is4Xkw=g,
ϕj+1-ϕjΔz=ik0n*2X[θϕj+1+(1-θ)ϕj],
ϕj+1=1+[(is)/2](1-θ)X1-[(is)/2]θXϕj.
1-isθ2Xϕj+1=1+is(1-θ)2Xϕj
1-isθ2X11-isθ2X2ϕj+11+is(1-θ)2X-s2θ24X1X2ϕj,
ϕj+11+is21-isθ2X2-11-isθ2X1-1Xϕj.
(X+β)w=g,
(X1+σm)w(m+1/2)=-(X2+β-σm)w(m)+g,(X2+ρm)w(m+1)=-(X1+β-ρm)w(m+1/2)+g.
w(m+1)=Mw(m)+v,
M=(X2+ρm)-1(X1+β-ρm)×(X1+σm)-1(X2+β-σm),v=(σm+ρm-β)(X2+ρm)-1(X1+σm)-1g.
(1-M)w=v.
Mˆ=X2+β-σX2+ρ·X1+β-ρX1+σ.
F(σ, ρ)=maxλ1,λ2λ2+β-σλ2+ρ·λ1+β-ρλ1+σ,
λlowλ10,    λlow=-4(k0n*Δx)2-12.
G(σ)=maxλlowλ0λ+β-σλ+σ.
x=±[1-cos(πz/L)]  µm

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