Abstract

We demonstrate the usefulness of the recently introduced modified Padé approximant operators for the solution of time-domain beam propagation problems. We show this both for a wideband method, which can take reflections into account, and for a split-step method for the modeling of ultrashort unidirectional pulses. The resulting approaches achieve high-order accuracy not only in space but also in time.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Yamauchi, Propagating Beam Analysis of Optical Waveguides (Research Studies Press, 2003).
  2. 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]
  3. 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]
  4. M. D. Feit and J. A. Fleck Jr., “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. Am. A 7, 73-79 (1990).
    [CrossRef]
  5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations,” IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  6. S. T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided wave optical structures,” J. Lightwave Technol. 7, 2033-2038 (1989).
    [CrossRef]
  7. J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
    [CrossRef]
  8. R. Y. Chan and J. M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett. 6, 1001-1003 (1994).
    [CrossRef]
  9. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits,” J. Lightwave Technol. 22, 684-691 (2004).
    [CrossRef]
  10. M. Koshiba, J. Tsuji, and M. Kikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102-110 (2000).
    [CrossRef]
  11. J. Shibayama, A. Yamahira, T. Mugita, J. Yamauchi, and H. Nakano, “A finite-difference time-domain beam-propagation method for TE- and TM-wave analyses,” J. Lightwave Technol. 21, 1709-1715 (2003).
    [CrossRef]
  12. P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
    [CrossRef]
  13. K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252-1254 (2009).
    [CrossRef]
  14. N. N. Feng, G. R. Zhou, and W. P. Huang, “An efficient split-step time-domain beam propagation method for modeling of optical waveguide devices,” J. Lightwave Technol. 23, 2186-2191 (2005).
    [CrossRef]
  15. H. M. Masoudi, “A novel nonparaxial time-domain beam propagation method for modeling ultrashort pulses in optical structures,” J. Lightwave Technol. 25, 1-10 (2007).
    [CrossRef]
  16. J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
    [CrossRef]
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

2009 (1)

K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252-1254 (2009).
[CrossRef]

2008 (1)

2007 (1)

H. M. Masoudi, “A novel nonparaxial time-domain beam propagation method for modeling ultrashort pulses in optical structures,” J. Lightwave Technol. 25, 1-10 (2007).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

2002 (1)

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

2000 (2)

M. Koshiba, J. Tsuji, and M. Kikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102-110 (2000).
[CrossRef]

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]

1996 (1)

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
[CrossRef]

1995 (1)

P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
[CrossRef]

1994 (1)

R. Y. Chan and J. M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett. 6, 1001-1003 (1994).
[CrossRef]

1990 (1)

1989 (1)

S. T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided wave optical structures,” J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations,” IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Aoki, S.

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
[CrossRef]

Benson, T. M.

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

Bienstman, P.

Chan, R. Y.

R. Y. Chan and J. M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett. 6, 1001-1003 (1994).
[CrossRef]

Chaudhuri, S.

S. T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided wave optical structures,” J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Choa, F. S.

P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
[CrossRef]

Chu, S. T.

S. T. Chu and S. Chaudhuri, “A finite-difference time-domain method for the design and analysis of guided wave optical structures,” J. Lightwave Technol. 7, 2033-2038 (1989).
[CrossRef]

Feit, M. D.

Feng, N. N.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

Fleck, J. A.

Fujisawa, T.

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]

Huang, W. P.

Kikari, M.

Koshiba, M.

Larkins, E. C.

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

Le, K. Q.

Lim, J. J.

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

Liu, J. M.

R. Y. Chan and J. M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett. 6, 1001-1003 (1994).
[CrossRef]

Liu, P. L.

P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
[CrossRef]

Masoudi, H. M.

H. M. Masoudi, “A novel nonparaxial time-domain beam propagation method for modeling ultrashort pulses in optical structures,” J. Lightwave Technol. 25, 1-10 (2007).
[CrossRef]

Mita, M.

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
[CrossRef]

Mugita, T.

Nakano, H.

J. Shibayama, A. Yamahira, T. Mugita, J. Yamauchi, and H. Nakano, “A finite-difference time-domain beam-propagation method for TE- and TM-wave analyses,” J. Lightwave Technol. 21, 1709-1715 (2003).
[CrossRef]

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (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]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

Rubio, R. G.

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.

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

Shibayama, J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

Tsuji, J.

Vettering, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

Yamahira, A.

Yamauchi, J.

J. Shibayama, A. Yamahira, T. Mugita, J. Yamauchi, and H. Nakano, “A finite-difference time-domain beam-propagation method for TE- and TM-wave analyses,” J. Lightwave Technol. 21, 1709-1715 (2003).
[CrossRef]

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
[CrossRef]

J. Yamauchi, Propagating Beam Analysis of Optical Waveguides (Research Studies Press, 2003).

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations,” IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

Zhao, Q.

P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
[CrossRef]

Zhou, G. R.

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. (3)

J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, “Analysis of antireflection coatings using the FD-TD method with the PML absorbing boundary condition,” IEEE Photon. Technol. Lett. 8, 239-241 (1996).
[CrossRef]

R. Y. Chan and J. M. Liu, “Time-domain wave propagation in optical structures,” IEEE Photon. Technol. Lett. 6, 1001-1003 (1994).
[CrossRef]

P. L. Liu, Q. Zhao, and F. S. Choa, “Slow-wave finite-difference beam propagation method,” IEEE Photon. Technol. Lett. 7, 890-892 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations,” IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).

J. Lightwave Technol. (6)

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

Microwave Opt. Technol. Lett. (1)

J. J. Lim, T. M. Benson, E. C. Larkins, and P. Sewell, “Wideband finite-difference-time-domain beam propagation method,” Microwave Opt. Technol. Lett. 34, 243-246 (2002).
[CrossRef]

Opt. Commun. (1)

K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252-1254 (2009).
[CrossRef]

Opt. Express (1)

Other (2)

J. Yamauchi, Propagating Beam Analysis of Optical Waveguides (Research Studies Press, 2003).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

The absolute values of 1 ( 1 X ) 1 2 , its first-order standard Padé approximant ( X 2 ) ( 1 X 4 ) and modified Padé approximant ( X 2 ) ( 1 X { 4 ( 1 + ibeta 2 ) } ) with respect to X.

Fig. 2
Fig. 2

The absolute values of 1 ( 1 X ) 1 2 (black line), its first-order standard Padé approximant ( X 2 ) ( 1 X 4 ) and modified Padé approximant ( X 2 ) ( 1 X { 4 ( 1 + ibeta 2 ) } ) with respect to X.

Fig. 3
Fig. 3

Optical grating with modulated refractive index with input pulse at time t = 0 superimposed.

Fig. 4
Fig. 4

Time evolution of the field monitored at the reference point.

Fig. 5
Fig. 5

Relative error of the field monitored at the reference point calculated by the modified gray curves (red online) and conventional black curves (blue online) Padé-based TD-BPM with various time steps using the field at 0.1 fs as a reference.

Fig. 6
Fig. 6

2D Y-branch waveguide.

Fig. 7
Fig. 7

Time evolution of transverse input field (a) and output fields after propagating 20 μ m calculated by TD-BPM based on the conventional (b) and the modified (c) Padé(1,1) approximant operator in the Y-branch waveguide. Each part of the figure shows the moving time window of width 120 fs used to monitor the pulse. The local waveguide geometry has been superimposed as a guide to the reader.

Tables (1)

Tables Icon

Table 1 Relative Error (%) of Ultrashort Pulses Calculated by the Modified and Conventional Padé-Based TD-BPM with Various Propagation Steps Using a Pulse Modeled with 0.02 μ m Step as a Reference

Equations (15)

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

2 Ψ z 2 + 2 Ψ x 2 = n 2 c 2 2 Ψ t 2 ,
Ψ ( x , z , t ) = ψ ( x , z , t ) exp ( i ω 0 t ) .
2 ψ t 2 + 2 i ω 0 ψ t = P ψ ,
ψ t = i ω 0 ( 1 1 X ) ψ ,
| t | n + 1 = i ω 0 X 2 1 j ω 0 | t | n .
ψ t i ω 0 X 2 1 X 4 ψ .
1 1 X X 2 1 X 4 .
ψ t i ω 0 X 2 1 X 4 ( 1 + i β 2 ) ψ .
Ψ ( x , z , t ) = ψ ( x , z , t ) exp ( i k z ) exp ( i ω 0 t ) ,
2 ψ z 2 + 2 i k ψ z + 2 ψ x 2 n 2 c 2 ( 2 ψ t 2 + 2 i ω 0 ψ t ) + k 0 2 ( n 2 n ref 2 ) ψ = 0 .
2 ψ z 2 + 2 i k ψ z = Q ψ ,
ψ ( x , z , t = 0 ) = ψ 0 ( x ) exp [ ( z z 0 w 0 ) 2 ] exp [ i k eff ( z z 0 ) ] ,
RE = [ | ψ p Δ t ψ p 0.1 fs | 2 | ψ p 0.1 fs | 2 ] 1 2 ,
ψ ( x , τ ) = ψ 0 ( x ) exp [ ( c τ c τ 0 c T ) 2 ]
RE = [ | ψ Δ z ψ 0.02 μ m | 2 d x d ( c τ ) | ψ 0.02 μ m | 2 d x d ( c τ ) ] 1 2 ,

Metrics