Abstract

Applying continuous and discrete transformation techniques, new analytical expressions to calculate dispersion and stability of a Runge-Kutta Finite Difference Beam Propagation Method (RK-FDBPM) are obtained. These expressions give immediate insight about the discretization errors introduced by the numerical method in the plane-wave spectrum domain. From these expressions a novel strategy to adequately set the mesh steps sizes of the RK-FDBPM is presented. Assessment of the method is performed by studying the propagation in several linear and nonlinear photonic devices for different spatial discretizations.

© 2008 Optical Society of America

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  1. A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, "Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides," J. Opt. Soc. Am. A 21, 2180-2185 (2004).
    [CrossRef]
  2. Z. Chen, Z. Li, and B. Li, "A 2-to-4 decoder switch in SiGe/Si multimode interference," Opt. Express 14, 2671-2678 (2006).
    [CrossRef] [PubMed]
  3. J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
    [CrossRef]
  4. D. Dai, J. He, and S. He, "Compact silicon-on-insulator-based multimode interference coupler with bilevel taper structure," Appl. Opt. 44, 5036-5041 (2005).
    [CrossRef] [PubMed]
  5. S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).
  6. M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003).
    [CrossRef]
  7. M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
    [CrossRef]
  8. N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
    [CrossRef]
  9. M. Takenaka and Y. Nakano, "Simulation of all-optical flip-flops based on bistable laser diodes with nonlinear couplers," in Proceedings of the 4th Int. Conf. on Numerical Simulation of Optoelectronic Devices (NUSOD ???04), 15-18 (2004).
  10. C. Ma and E. Van Keuren, "A three-dimensional wide-angle BPM for optical waveguide structures," Opt. Express 15, 402-407 (2007).
    [CrossRef] [PubMed]
  11. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
    [CrossRef]
  12. S. Sujecki "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. A 25, 138-145 (2008).
    [CrossRef]
  13. J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004).
    [CrossRef]
  14. J. de-Oliva-Rubio, Desarrollo y validación de técnicas de diferencias finitas para el análisis de dispositivos ópticos lineales y no-lineales, Ph. D. Thesis (in Spanish), ISBN: 84-690-3321-2, (Universidad de Málaga, 2006).
  15. W. P. Huang and 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]
  16. M. Matsuhara, "A novel beam propagation method based on the Galerkin method," Electron. Commun. Jpn. Pt. II 73, 41-47 (1990).
    [CrossRef]
  17. B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, 2nd Ed. (John Wiley & Sons, 2007), Chap. 4, pp. 105-112.
  18. I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
    [CrossRef]
  19. C. Vassallo, Optical waveguides concepts, ser. Optical waveguides sciences and technology (Elsevier, Amsterdam, 1991), Chap. 1, pp. 18-26.
  20. I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
    [CrossRef]
  21. K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
    [CrossRef]

2008

2007

2006

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Z. Chen, Z. Li, and B. Li, "A 2-to-4 decoder switch in SiGe/Si multimode interference," Opt. Express 14, 2671-2678 (2006).
[CrossRef] [PubMed]

J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
[CrossRef]

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

2005

2004

J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004).
[CrossRef]

A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, "Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides," J. Opt. Soc. Am. A 21, 2180-2185 (2004).
[CrossRef]

2003

M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003).
[CrossRef]

1994

I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
[CrossRef]

1993

W. P. Huang and 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]

1990

M. Matsuhara, "A novel beam propagation method based on the Galerkin method," Electron. Commun. Jpn. Pt. II 73, 41-47 (1990).
[CrossRef]

K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
[CrossRef]

Bao, L.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Barton, J. S.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

Bosisio, R. G.

I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
[CrossRef]

Camacho Peñalosa, C.

I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
[CrossRef]

Cassan, E.

Chen, Z.

Dai, D.

de-Oliva-Rubio, J.

J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004).
[CrossRef]

Elkin, N. N.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Gardes, F. Y.

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

Hagness, S. C.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Hayata, K.

K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
[CrossRef]

He, J.

He, S.

Huang, W. P.

W. P. Huang and 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]

Kim, N.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Koshiba, M.

K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
[CrossRef]

Koster, A.

Laval, S.

Lee, S. T.

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

Lee, T.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Li, B.

Li, Z.

Ma, C.

Matsuhara, M.

M. Matsuhara, "A novel beam propagation method based on the Galerkin method," Electron. Commun. Jpn. Pt. II 73, 41-47 (1990).
[CrossRef]

Mawst, J.L

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Misawa, A.

K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
[CrossRef]

Molina Fernández, I.

I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
[CrossRef]

Molina-Fernández, I.

J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004).
[CrossRef]

I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
[CrossRef]

Nakano, H.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
[CrossRef]

Nakano, Y.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003).
[CrossRef]

Napartovich, A. P.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Ortega-Moñux, A.

I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
[CrossRef]

Pascal, D.

Png, C. E.

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

Raburn, M.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

Ramos, J. I.

I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
[CrossRef]

Reed, G. T.

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

Shibayama, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

Song, X.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

Sujecki, S.

Sumida, K.

J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
[CrossRef]

Takahashi, T.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

Takeda, K.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

Takenaka, M.

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003).
[CrossRef]

Troschieva, V. N.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Van Keuren, E.

Vivien, L.

Vysotsky, D. V.

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

Wangüemert-Pérez, J. G.

I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
[CrossRef]

Xu, C. L.

W. P. Huang and 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.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
[CrossRef]

Appl. Opt.

Electron. Commun. Jpn

M. Matsuhara, "A novel beam propagation method based on the Galerkin method," Electron. Commun. Jpn. Pt. II 73, 41-47 (1990).
[CrossRef]

IEEE J. Lightwave Technol.

S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).

N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006).
[CrossRef]

IEEE J. Quantum Electron.

W. P. Huang and 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]

IEEE Photon. Technol. Lett.

M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003).
[CrossRef]

M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006).
[CrossRef]

J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006).
[CrossRef]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994).
[CrossRef]

J Opt Soc Am B

K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004).
[CrossRef]

Opt. Express

Other

M. Takenaka and Y. Nakano, "Simulation of all-optical flip-flops based on bistable laser diodes with nonlinear couplers," in Proceedings of the 4th Int. Conf. on Numerical Simulation of Optoelectronic Devices (NUSOD ???04), 15-18 (2004).

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, 2nd Ed. (John Wiley & Sons, 2007), Chap. 4, pp. 105-112.

J. de-Oliva-Rubio, Desarrollo y validación de técnicas de diferencias finitas para el análisis de dispositivos ópticos lineales y no-lineales, Ph. D. Thesis (in Spanish), ISBN: 84-690-3321-2, (Universidad de Málaga, 2006).

C. Vassallo, Optical waveguides concepts, ser. Optical waveguides sciences and technology (Elsevier, Amsterdam, 1991), Chap. 1, pp. 18-26.

I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005)
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Normalized phase error for the different meshes. (a)-mesh Δy=0.5 µm, Δz=0.1 λ0. (b)-mesh Δy=0.1 µm, Δz=0.1 λ0. (c)-mesh Δy=0.025 µm, Δz=0.0125 λ0. The superimposed black line is the normalized envelope spatial spectrum.

Fig. 2.
Fig. 2.

Propagation of the TE0 excitation in the tilted slab. In Fig. 2(b) the (b)-mesh and the (c)-mesh solutions are indistinguishable.

Fig. 3.
Fig. 3.

Propagating and fundamental mode fields overlap integral: S21. Fig. 3(a): S21 amplitude. Fig. 3(b): S21 phase error regarding fundamental mode phase at d=z/cosθ. The blue line corresponds to the (a)-mesh, the red line to the (b)-mesh and the green line to the (c)-mesh.

Fig. 4.
Fig. 4.

MMI 2×3 coupler geometry

Fig. 5.
Fig. 5.

Variations in RK-FDBPM phase error depending on mesh size. nref=1,4487, n=1.4549, λ0=1.55µm, L=3570µm.

Fig. 6.
Fig. 6.

Longitudinal distribution of the electric field for nref=1,4487, λ0=1.55µm. (a) Δx=Δy=1µm, Δz=0.1 µm. (b) Δx=Δy=0.1µm., Δz=0.01 µm.

Fig. 7.
Fig. 7.

Numerical analysis results for the non-linear slab characterization. (a) Variation of phase error for different mesh step sizes. (b) Variation with Δz of the numerical attenuation constant for the mesh with Δy=0.25 µm. For those Δz with positive attenuation in the bandwidth ends (red and blue curves in (b)), the method becomes unstable.

Fig. 8.
Fig. 8.

Electric field in the non-linear slab with Δy=0.5 µm and Δz=8λ0. It can be seen how two solitons are generated at z=400λ0 y z=1200λ0. They propagate in the non-linear substrate (y<-10µm).

Fig. 9.
Fig. 9.

Field in the non-linear slab at z=2400λ0 for the different proposed meshes. Field strength is normalized with respect to its maximum value in the structure.

Equations (35)

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j e x z = M x x e x + M x y e y , j E y z = M y x e x + M y y e y ,
ε ¯ ( x , y , z , t ) = e ¯ ( x , y , z ) e j ( ω t k 0 n ref z ) ,
j e u z = M u u e u ,
M uu e u = 1 2 β 0 { u [ 1 n 2 u ( n 2 e u ) ] + 2 v 2 e u + ( n 2 n ref 2 ) k 0 2 e u } ,
d Ψ ¯ ( z ) d z = M ̿ T ( z ) · Ψ ¯ ( z ) ,
Ψ ¯ p + 1 = Ψ ¯ p + Δ z · M ̿ T · Ψ ¯ p + Δ z 2 2 · M ̿ T 2 · Ψ ¯ p + Δ z 3 6 · M ̿ T 3 · Ψ ¯ p + Δ z 4 24 · M ̿ T 4 · Ψ ¯ p .
d i Ψ ¯ d z i = M ̿ T i · Ψ ¯ ,
e u z = 1 2 j k 0 n ref [ 2 x 2 e u + 2 y 2 e u + k 0 ( n 2 n ref 2 ) e u ] .
E u ( β x , β y , β z ) = x = y = z = e u ( x , y , z ) e j ( β x x + β y y + β z z ) dxdydz ,
E u ( β x , β y , β z ) D u ( β x , β y , β z ) = 0 .
D u ( β x , β y , β z ) = 0 β z = 1 2 j k 0 n ref [ β x 2 + β y 2 + k 0 ( n 2 n ref 2 ) ] .
β z = 1 2 j k 0 n ref [ β y 2 + k 0 ( n 2 n ref 2 ) ] .
E u ( X , Y , Z ) = m = n = p = e u m , n , p X m Y n Z p .
X = e j Ω x , Y = e j Ω y , Z = e ( A z + j Ω z ) ,
β x = Ω x Δ x , β y = Ω y Δ y , γ z RK = α z RK + j β z RK = A z + j Ω z Δ z ,
γ z RK = 1 Δ z L n [ ( 1 + Δ z M ̿ T , 0 + Δ z 2 2 M ̿ T , 0 2 + Δ z 3 6 M ̿ T , 0 3 + Δ z 4 24 M ̿ T , 0 4 )
            + ( 2 Δ z M ̿ T , 1 + Δ z 2 M ̿ T , 1 2 + Δ z 3 3 M ̿ T , 1 3 + Δ z 4 12 M ̿ T , 1 4 ) cos ( β x Δ x )
              + ( 2 Δ z M ̿ T , N x + Δ z 2 M ̿ T , N x 2 + Δ z 3 3 M ̿ T , N x 3 + Δ z 4 12 M ̿ T , N x 4 ) cos ( β y Δ y )
              + ( Δ z 2 M ̿ T , 2 2 + Δ z 3 3 M ̿ T , 2 3 + Δ z 4 12 M ̿ T , 2 4 ) cos ( 2 β x Δ x )
                + ( Δ z 2 M ̿ T , 2 N x 2 + Δ z 3 3 M ̿ T , 2 N x 3 + Δ z 4 12 M ̿ T , 2 N x 4 ) cos ( 2 β y Δy )
              + ( Δ z 3 3 M ̿ T , 3 3 + Δ z 4 12 M ̿ T , 3 4 ) cos ( 3 β x Δ x ) + ( Δ z 3 3 M ̿ T , 3 N x 3 + Δ z 4 12 M ̿ T , 3 N x 4 ) cos ( 3 β y Δ y )
                  + Δ z 4 12 M ̿ T , 4 4 cos ( 4 β x Δ x ) + Δ z 4 12 M ̿ T , 4 N x 4 cos ( 4 β y Δ y )
                  + ( 2 Δ z 2 M ̿ T , N x + 1 2 + 2 Δ z 3 3 M ̿ T , N x + 1 3 + Δ z 4 6 M ̿ T , N x + 1 4 ) cos ( β x Δ x ) cos ( β y Δ y )
                  + ( 2 Δ z 3 3 M ̿ T , N x + 2 3 + Δ z 4 6 M ̿ T , N x + 2 4 ) cos ( 2 β x Δ x ) cos ( β y Δ y )
                + ( 2 Δ z 3 3 M ̿ T , 2 N x + 1 3 + Δ z 4 6 M ̿ T , 2 N x + 1 4 ) cos ( β x Δ x ) cos ( 2 β y Δ y ) +
                    + Δ z 4 6 M ̿ T , N x + 3 4 cos ( 3 β x Δ x ) cos ( β y Δ y ) + Δ z 4 6 M ̿ T , 3 N x + 1 4 cos ( β x Δ x ) cos ( 3 β y Δ y )
                    + Δ z 4 6 M ̿ T , 2 N x + 2 4 cos ( 4 β x Δ x ) cos ( 4 β y Δ y ) ] ,
γ z RK = 1 Δ z L n [ ( 1 + Δ z M ̿ T , 0 + Δ z 2 2 M ̿ T , 0 2 + Δ z 3 6 M ̿ T , 0 3 + Δ z 4 24 M ̿ T , 0 4 )
              + ( 2 Δ z M ̿ T , 1 + Δ z 2 M ̿ T , 1 2 + Δ z 3 3 M ̿ T , 1 3 + Δ z 4 12 M ̿ T , 1 4 ) cos ( β y Δy )
              + ( Δ z 2 M ̿ T , 2 2 + Δ z 3 3 M ̿ T , 2 3 + Δ z 4 12 M ̿ T , 2 4 ) cos ( 2 β y Δy )
                + ( Δ z 3 3 M ̿ T , 3 3 + Δ z 4 12 M ̿ T , 3 4 ) cos ( 3 β y Δ y ) + ( Δ z 4 12 M ̿ T , 4 4 ) cos ( 4 β y Δ y ) ] .
Δ β z L = β z β z RK L .
Δ β z L 2 π 1 ,
P ̿ Ψ ¯ p + 1 = Q ̿ · Ψ ¯ p ,
S 3 , 1 = S 5 , 2 = 1 3 e j ϕ , S 4 , 1 = S 4 , 2 = 1 3 e j ( ϕ π ) , S 4 , 1 = S 4 , 2 = 1 3 e j ( ϕ 2 π 3 ) .

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