Abstract

We propose a general method for analyzing a multilayer optical waveguide with all nonlinear layers. This general method can be degenerated into some special cases, such as symmetric or asymmetric nonlinear optical waveguide structures, the intensity-dependent refractive index with self-focusing nonlinear medium, hollow waveguides, and multilayer systems. Based on this general method, the analysis and calculation of complicated multilayer optical planar waveguides can be achieved easily. The analytical and numerical results show excellent agreement.

© 2007 Optical Society of America

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  1. K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
    [CrossRef]
  2. Y. D. Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005-4012 (2006).
    [CrossRef] [PubMed]
  3. Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum Electron. 11, 307-312 (2005).
    [CrossRef]
  4. J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995).
    [CrossRef]
  5. T. Rozzi, "Modal analysis of nonlinear propagation in dielectric slab waveguide," J. Lightwave Technol. 14, 229-234 (1996).
    [CrossRef]
  6. R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
    [CrossRef]
  7. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
    [CrossRef]
  8. G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
    [CrossRef]
  9. J. G. Ma and I. Wolff, "TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media," IEEE Trans. Microw. Theory Tech. 44, 730-738 (1996).
    [CrossRef]
  10. A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319-324 (1986).
    [CrossRef]
  11. A. D. Boardman and P. Egan, "S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media," IEEE J. Quantum Electron. 21, 1701-1713 (1985).
    [CrossRef]
  12. S. W. Kang, "Optical slab waveguides of Kerr material with linear profile," Opt. Commun. 174, 127-131 (2000).
    [CrossRef]
  13. W. Chen and A. A. Maradudin, "S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film," J. Opt. Soc. Am. B 5, 529-538 (1988).
    [CrossRef]
  14. Y. F. Li and K. Iizuka, "Unified nonlinear waveguide dispersion equation without spurious roots," IEEE J. Quantum Electron. 31, 791-794 (1995).
    [CrossRef]
  15. S. Okafuji, "Propagation characteristics of TE0 waves in three-layer optical waveguides with self focusing and self defocusing nonlinear layers," IEEE Proceedings-J 140, 127-136 (1993).
  16. R. A. Sammut, C. Pask, and Q. Y. Li, "Theoretical study of spatial solitons in planar waveguides," J. Opt. Soc. Am. B 10, 485-491 (1993).
    [CrossRef]
  17. K. S. Chiang and R. A. Sammut, "Effective-index method for spatial solitons in planar waveguides with Kerr-type nonlinearity," J. Opt. Soc. Am. B 10, 704-708 (1993).
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    [CrossRef]
  19. T. T. Shi and S. Chi, "Nonlinear TE-wave propagation in a symmetric, converging, single-mode Y-junction waveguide," J. Opt. Soc. Am. B 9, 1338-1340 (1992).
    [CrossRef]
  20. G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
    [CrossRef]
  21. S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
    [CrossRef]
  22. T. Sakakibara and N. Okamoto, "Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers," IEEE J. Quantum Electron. 23, 2084-2088 (1987).
    [CrossRef]
  23. J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
    [CrossRef]
  24. S. Radic, N. George, and G. P. Agrawal, "Optical switching in λ/4-shifted nonlinear periodic structures," Opt. Lett. 19, 1789-1791 (1994).
    [CrossRef] [PubMed]
  25. S. Radic, N. George, and G. P. Agrawal, "Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures," J. Opt. Soc. Am. B 12, 671-680 (1995).
    [CrossRef]
  26. S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995)
    [CrossRef]
  27. U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
    [CrossRef]
  28. K. Ogusu, "Analysis of nonlinear multilayer waveguides with Kerr-like permittivities," Opt. Quantum Electron. 21, 109-116 (1989).
    [CrossRef]
  29. S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding," Opt. Commun. 161, 141-148 (1999).
    [CrossRef]
  30. M. H. Chen and Y. D. Wu, "A general method for analyzing the multi-layer planar waveguide," Fiber Integr. Opt. 11, 159-176 (1992).
    [CrossRef]
  31. Y. D. Wu, M. H. Chen, and H. J. Tasi, "Analyzing multilayer optical waveguides with nonlinear cladding and substrates," J. Opt. Soc. Am. B 19, 1737-1745 (2002).
    [CrossRef]
  32. Y. D. Wu, "Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film," IEEE J. Quantum Electron. 40, 529-540 (2004).
    [CrossRef]
  33. Y. D. Wu and M. H. Chen, "Method for analyzing multilayer nonlinear optical waveguide," Opt. Express 13, 7982-7995 (2005).
    [CrossRef] [PubMed]
  34. N. Saiga, "Calculation of TE modes in graded-index nonlinear optical waveguides with arbitrary profile of refractive index," J. Opt. Soc. Am. B 8, 88-94 (1991).
    [CrossRef]
  35. Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber Integr. Opt. 23, 387-404 (2004).
    [CrossRef]
  36. Y. D. Wu, "Coupled-soliton all-optical logic device with two parallel tapered waveguides," Fiber Integr. Opt. 23, 405-414 (2004).
    [CrossRef]
  37. Y. D. Wu, "New all-optical wavelength auto-router based on spatial solitons," Opt. Express 12, 4172-4177 (2004).
    [CrossRef] [PubMed]
  38. W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
    [CrossRef]
  39. W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. 11, 32-39 (1975).
    [CrossRef]
  40. M. Izutsu, Y. Nakai, and T. Sueta, "Operation mechanism of the single-mode optical-waveguide Y junction," Opt. Lett. 7, 136-138 (1982).
    [CrossRef] [PubMed]
  41. M. O. Twati and T. J. F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327-332 (2002).
    [CrossRef]
  42. H. Yokota, K. Kimura, and S. Kurazono, "Numerical analysis of an optical X coupler with a nonlinear dieletric region," IEICE Trans. Electron. E78-C, 61-66 (1995).
  43. Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
    [CrossRef]
  44. G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
    [CrossRef]

2006

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Y. D. Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005-4012 (2006).
[CrossRef] [PubMed]

2005

Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum Electron. 11, 307-312 (2005).
[CrossRef]

Y. D. Wu and M. H. Chen, "Method for analyzing multilayer nonlinear optical waveguide," Opt. Express 13, 7982-7995 (2005).
[CrossRef] [PubMed]

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

2004

Y. D. Wu, "Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film," IEEE J. Quantum Electron. 40, 529-540 (2004).
[CrossRef]

Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber Integr. Opt. 23, 387-404 (2004).
[CrossRef]

Y. D. Wu, "Coupled-soliton all-optical logic device with two parallel tapered waveguides," Fiber Integr. Opt. 23, 405-414 (2004).
[CrossRef]

Y. D. Wu, "New all-optical wavelength auto-router based on spatial solitons," Opt. Express 12, 4172-4177 (2004).
[CrossRef] [PubMed]

2002

Y. D. Wu, M. H. Chen, and H. J. Tasi, "Analyzing multilayer optical waveguides with nonlinear cladding and substrates," J. Opt. Soc. Am. B 19, 1737-1745 (2002).
[CrossRef]

M. O. Twati and T. J. F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327-332 (2002).
[CrossRef]

2000

G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
[CrossRef]

S. W. Kang, "Optical slab waveguides of Kerr material with linear profile," Opt. Commun. 174, 127-131 (2000).
[CrossRef]

1999

S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding," Opt. Commun. 161, 141-148 (1999).
[CrossRef]

1998

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

1996

J. G. Ma and I. Wolff, "TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media," IEEE Trans. Microw. Theory Tech. 44, 730-738 (1996).
[CrossRef]

T. Rozzi, "Modal analysis of nonlinear propagation in dielectric slab waveguide," J. Lightwave Technol. 14, 229-234 (1996).
[CrossRef]

1995

J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures," J. Opt. Soc. Am. B 12, 671-680 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995)
[CrossRef]

J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995).
[CrossRef]

Y. F. Li and K. Iizuka, "Unified nonlinear waveguide dispersion equation without spurious roots," IEEE J. Quantum Electron. 31, 791-794 (1995).
[CrossRef]

H. W. Schürmann," On the theory of TE-polarized waves guided by a nonlinear three-layer structure," Z. Phys. B 97, 515-522 (1995).
[CrossRef]

1994

1993

1992

T. T. Shi and S. Chi, "Nonlinear TE-wave propagation in a symmetric, converging, single-mode Y-junction waveguide," J. Opt. Soc. Am. B 9, 1338-1340 (1992).
[CrossRef]

M. H. Chen and Y. D. Wu, "A general method for analyzing the multi-layer planar waveguide," Fiber Integr. Opt. 11, 159-176 (1992).
[CrossRef]

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

1991

1989

U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
[CrossRef]

K. Ogusu, "Analysis of nonlinear multilayer waveguides with Kerr-like permittivities," Opt. Quantum Electron. 21, 109-116 (1989).
[CrossRef]

1988

W. Chen and A. A. Maradudin, "S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film," J. Opt. Soc. Am. B 5, 529-538 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
[CrossRef]

1987

T. Sakakibara and N. Okamoto, "Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers," IEEE J. Quantum Electron. 23, 2084-2088 (1987).
[CrossRef]

1986

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319-324 (1986).
[CrossRef]

1985

A. D. Boardman and P. Egan, "S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media," IEEE J. Quantum Electron. 21, 1701-1713 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

1984

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

1982

1975

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. 11, 32-39 (1975).
[CrossRef]

Agrawal, G. P.

Binder, R.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

Boardman, A. D.

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319-324 (1986).
[CrossRef]

A. D. Boardman and P. Egan, "S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media," IEEE J. Quantum Electron. 21, 1701-1713 (1985).
[CrossRef]

Burak, D.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

Burns, W. K.

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. 11, 32-39 (1975).
[CrossRef]

Chan, Y. K.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

Chaudhuri, S. K.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Chen, M. H.

Chen, W.

Chi, S.

Chiang, K. S.

Chien, S. F.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

Chilwell, J.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

Christodoulides, D. N.

G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
[CrossRef]

Chu, S. T.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Egan, P.

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319-324 (1986).
[CrossRef]

A. D. Boardman and P. Egan, "S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media," IEEE J. Quantum Electron. 21, 1701-1713 (1985).
[CrossRef]

Finalyson, N.

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

Fukuda, H.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

George, N.

Golz, M.

U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
[CrossRef]

Huang, W.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Iizuka, K.

Y. F. Li and K. Iizuka, "Unified nonlinear waveguide dispersion equation without spurious roots," IEEE J. Quantum Electron. 31, 791-794 (1995).
[CrossRef]

Itabashi, S.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Izutsu, M.

Jeong, J. S.

J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
[CrossRef]

Kang, S. W.

S. W. Kang, "Optical slab waveguides of Kerr material with linear profile," Opt. Commun. 174, 127-131 (2000).
[CrossRef]

Kivshar, Y. S.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

Kwak, C. H.

J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
[CrossRef]

Lederer, F.

U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
[CrossRef]

Lee, E. H.

J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
[CrossRef]

Li, Q. Y.

Li, Y. F.

Y. F. Li and K. Iizuka, "Unified nonlinear waveguide dispersion equation without spurious roots," IEEE J. Quantum Electron. 31, 791-794 (1995).
[CrossRef]

Love, J. D.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

Low, A. L.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

Ma, J. G.

J. G. Ma and I. Wolff, "TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media," IEEE Trans. Microw. Theory Tech. 44, 730-738 (1996).
[CrossRef]

Ma, J.-G.

J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995).
[CrossRef]

Maradudin, A. A.

Micallef, R. W.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
[CrossRef]

Milton, A. F.

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. 11, 32-39 (1975).
[CrossRef]

Nakai, Y.

Ogusu, K.

K. Ogusu, "Analysis of nonlinear multilayer waveguides with Kerr-like permittivities," Opt. Quantum Electron. 21, 109-116 (1989).
[CrossRef]

Okafuji, S.

S. Okafuji, "Propagation characteristics of TE0 waves in three-layer optical waveguides with self focusing and self defocusing nonlinear layers," IEEE Proceedings-J 140, 127-136 (1993).

Okamoto, N.

T. Sakakibara and N. Okamoto, "Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers," IEEE J. Quantum Electron. 23, 2084-2088 (1987).
[CrossRef]

Pask, C.

Pavlasek, T. J. F

M. O. Twati and T. J. F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327-332 (2002).
[CrossRef]

Radic, S.

Rozzi, T.

T. Rozzi, "Modal analysis of nonlinear propagation in dielectric slab waveguide," J. Lightwave Technol. 14, 229-234 (1996).
[CrossRef]

Saiga, N.

Sakakibara, T.

T. Sakakibara and N. Okamoto, "Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers," IEEE J. Quantum Electron. 23, 2084-2088 (1987).
[CrossRef]

Sammut, R. A.

Schürmann, H. W.

H. W. Schürmann," On the theory of TE-polarized waves guided by a nonlinear three-layer structure," Z. Phys. B 97, 515-522 (1995).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

Segev, M.

G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
[CrossRef]

She, S.

S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding," Opt. Commun. 161, 141-148 (1999).
[CrossRef]

Shi, T. T.

Shin, S. Y.

S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
[CrossRef]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

Shoji, T.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
[CrossRef]

S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

Sueta, T.

Tasi, H. J.

Tsuchizawa, T.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Ttrutschel, U.

U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
[CrossRef]

Twati, M. O.

M. O. Twati and T. J. F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327-332 (2002).
[CrossRef]

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

Watanabe, T.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Wolff, I.

J. G. Ma and I. Wolff, "TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media," IEEE Trans. Microw. Theory Tech. 44, 730-738 (1996).
[CrossRef]

J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995).
[CrossRef]

Wong, H. Y.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
[CrossRef]

Wu, Y. D.

Y. D. Wu, "New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005-4012 (2006).
[CrossRef] [PubMed]

Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum Electron. 11, 307-312 (2005).
[CrossRef]

Y. D. Wu and M. H. Chen, "Method for analyzing multilayer nonlinear optical waveguide," Opt. Express 13, 7982-7995 (2005).
[CrossRef] [PubMed]

Y. D. Wu, "Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film," IEEE J. Quantum Electron. 40, 529-540 (2004).
[CrossRef]

Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber Integr. Opt. 23, 387-404 (2004).
[CrossRef]

Y. D. Wu, "Coupled-soliton all-optical logic device with two parallel tapered waveguides," Fiber Integr. Opt. 23, 405-414 (2004).
[CrossRef]

Y. D. Wu, "New all-optical wavelength auto-router based on spatial solitons," Opt. Express 12, 4172-4177 (2004).
[CrossRef] [PubMed]

Y. D. Wu, M. H. Chen, and H. J. Tasi, "Analyzing multilayer optical waveguides with nonlinear cladding and substrates," J. Opt. Soc. Am. B 19, 1737-1745 (2002).
[CrossRef]

M. H. Chen and Y. D. Wu, "A general method for analyzing the multi-layer planar waveguide," Fiber Integr. Opt. 11, 159-176 (1992).
[CrossRef]

Xu, C.

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Yamada, K.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

Yong, Y. S.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

You, A. H.

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

Zhang, S.

S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding," Opt. Commun. 161, 141-148 (1999).
[CrossRef]

Appl. Phys. Lett.

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, "Nonlinear waves guided by thin films," Appl. Phys. Lett. 44, 830-832 (1984).
[CrossRef]

Fiber Integr. Opt.

M. H. Chen and Y. D. Wu, "A general method for analyzing the multi-layer planar waveguide," Fiber Integr. Opt. 11, 159-176 (1992).
[CrossRef]

Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber Integr. Opt. 23, 387-404 (2004).
[CrossRef]

Y. D. Wu, "Coupled-soliton all-optical logic device with two parallel tapered waveguides," Fiber Integr. Opt. 23, 405-414 (2004).
[CrossRef]

IEEE J. Quantum Electron.

Y. D. Wu, "Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film," IEEE J. Quantum Electron. 40, 529-540 (2004).
[CrossRef]

T. Sakakibara and N. Okamoto, "Nonlinear TE waves in a dielectric slab waveguide with two optically nonlinear layers," IEEE J. Quantum Electron. 23, 2084-2088 (1987).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, "Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method," IEEE J. Quantum Electron. 31, 1326-1336 (1995)
[CrossRef]

U. Ttrutschel, F. Lederer, and M. Golz, "Nonlinear guided waves in multilayer systems," IEEE J. Quantum Electron. 25, 194-200 (1989).
[CrossRef]

A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319-324 (1986).
[CrossRef]

A. D. Boardman and P. Egan, "S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media," IEEE J. Quantum Electron. 21, 1701-1713 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774-783 (1985).
[CrossRef]

Y. F. Li and K. Iizuka, "Unified nonlinear waveguide dispersion equation without spurious roots," IEEE J. Quantum Electron. 31, 791-794 (1995).
[CrossRef]

W. K. Burns and A. F. Milton, "Mode conversion in planar-dielectric separating waveguides," IEEE J. Quantum Electron. 11, 32-39 (1975).
[CrossRef]

Y. S. Yong, A. L. Low, S. F. Chien, A. H. You, H. Y. Wong, and Y. K. Chan, "Design and analysis of equal power divider using 4-branch waveguide," IEEE J. Quantum Electron. 41, 1181-1187 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

G. I. Stegeman and D. N. Christodoulides, and M. Segev, "Optical spatial solitons: historical perspectives," IEEE J. Sel. Top. Quantum Electron. 6, 1419-1427 (2000).
[CrossRef]

Y. D. Wu, "All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity," IEEE J. Sel. Top. Quantum Electron. 11, 307-312 (2005).
[CrossRef]

IEEE Photonics Technol. Lett.

K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanabe, T. Shoji, and S. Itabashi, "All-optical efficient wavelength conversion using silicon photonic wire waveguide," IEEE Photonics Technol. Lett. 18, 1046-1048 (2006).
[CrossRef]

IEEE Proceedings-J

S. Okafuji, "Propagation characteristics of TE0 waves in three-layer optical waveguides with self focusing and self defocusing nonlinear layers," IEEE Proceedings-J 140, 127-136 (1993).

IEEE Trans. Microw. Theory Tech.

J. G. Ma and I. Wolff, "TE wave properties of slab dielectric guide bounded by nonlinear non-Kerr-like media," IEEE Trans. Microw. Theory Tech. 44, 730-738 (1996).
[CrossRef]

J.-G. Ma and I. Wolff, "Propagation characteristics of TE-wave guided by thin films bounded by nonlinear media," IEEE Trans. Microw. Theory Tech. 43, 790-795 (1995).
[CrossRef]

J. Lightwave Technol.

T. Rozzi, "Modal analysis of nonlinear propagation in dielectric slab waveguide," J. Lightwave Technol. 14, 229-234 (1996).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finalyson, R. Zanoni, and C. T. Seaton, "Third order nonlinear integrated optics," J. Lightwave Technol. 6, 953-970 (1988).
[CrossRef]

S. Y. Shin, E. M. Wright, and G. I. Stegeman, "Nonlinear TE waves of coupled waveguides bounded by nonlinear media," J. Lightwave Technol. 6, 977-983 (1988).
[CrossRef]

W. Huang, C. Xu, S. T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

S. W. Kang, "Optical slab waveguides of Kerr material with linear profile," Opt. Commun. 174, 127-131 (2000).
[CrossRef]

S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear caldding," Opt. Commun. 161, 141-148 (1999).
[CrossRef]

J. S. Jeong, C. H. Kwak, and E. H. Lee, "Optical properties of TE nonlinear waves in planar waveguides with two nonlinear bounding layers," Opt. Commun. 116, 351-357 (1995).
[CrossRef]

M. O. Twati and T. J. F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327-332 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

K. Ogusu, "Analysis of nonlinear multilayer waveguides with Kerr-like permittivities," Opt. Quantum Electron. 21, 109-116 (1989).
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R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, "Generation of spatial solitons using non-linear guided modes," Opt. Quantum Electron. 30, 751-770 (1998).
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Z. Phys. B

H. W. Schürmann," On the theory of TE-polarized waves guided by a nonlinear three-layer structure," Z. Phys. B 97, 515-522 (1995).
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Cited By

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Figures (18)

Fig. 1
Fig. 1

The structure of multilayer optical waveguides with all nonlinear layers.

Fig. 2.
Fig. 2.

Diagram of the computation steps.

Fig. 3.
Fig. 3.

(a) Dispersion curve of the seven-layer optical waveguide structure with uniform nonlinearity. (b) The electric field distributions with respect to point A-D as shown in (a).

Fig. 4.
Fig. 4.

The multibranch optical waveguide structure.

Fig. 5.
Fig. 5.

The two-branch optical waveguide structure with all nonlinear layers.

Fig. 6.
Fig. 6.

For the low input power density case, electric-field distributions of the two-branch optical waveguide structure with all nonlinear layers at positions (a) Z1 (b)Z2 (c) Z3 (d) Z4.

Fig. 7.
Fig. 7.

For the high input power density case, electric-field distributions of the two-branch optical waveguide structure with all nonlinear layers at positions (a) Z1 (b)Z2 (c) Z3 (d) Z4.

Fig. 8.
Fig. 8.

The three-branch optical waveguide structure with uniform nonlinearity.

Fig. 9.
Fig. 9.

For the low input power density case electric-field distributions of the three-branch optical waveguide structure with all nonlinear layers at positions (a) Z1 (b)Z2 (c) Z3 (d) Z4.

Fig. 10.
Fig. 10.

For the high input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear layers at positions (a) Z1 (b)Z2 (c) Z3 (d) Z4.

Fig. 11.
Fig. 11.

The typical evolution of a wave propagating along a two-branch optical waveguide structure with all nonlinear layers at the low input power density.

Fig. 12.
Fig. 12.

The typical evolution of a wave propagating along a two-branch optical waveguide structure with all nonlinear layers at the high input power density.

Fig. 13.
Fig. 13.

The relevant curves shown in Fig. 6 and 11 on the same graph (a) at position Z1, (b) at position Z2, (;) at position Z3, (d) at position Z4, (dotted line: the predicted results, dashed line: the numerical simulation results).

Fig. 14.
Fig. 14.

The relevant curves shown in Fig. 7 and 12 on the same graph (a) at position Z1, (b) at position Z2, (c) at position Z3, (d) at position Z4, (dotted line: the predicted results, dashed line: the numerical simulation results).

Fig. 15.
Fig. 15.

The typical evolution of a wave propagating along a three-branch optical waveguide structure with uniform nonlinearity.

Fig. 16.
Fig. 16.

The typical evolution of a wave propagating along a three-branch optical waveguide structure with uniform nonlinearity.

Fig. 17.
Fig. 17.

The relevant curves shown in Fig. 9 and 15 on the same graph (a) at position Z1, (b) at position Z2, (c) at position Z3, (d) at position Z4, (dotted line: the predicted results, dashed line: the numerical simulation results).

Fig. 18.
Fig. 18.

The relevant curves shown in Fig. 10 and 16 on the same graph (a) at position Z1, (b) at position Z2, (c) at position Z3, (d) at position Z4, (dotted line: the predicted results, dashed line: the numerical simulation results).

Equations (70)

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

2 ψ yi = n i 2 c 2 2 ψ yi t 2 , i = 0,1,2 , . , M 1
ψ yi ( x , z , t ) = E i ( x ) exp [ j ( ωt n e k 0 z ) ] , x i x x i + 1
n i 2 = n i 0 2 + α i E i ( x ) 2 , i = 0,1,2 , . M 1
d 2 E i ( x ) d x 2 Q i 2 E i ( x ) + k 0 2 α i E i 3 ( x ) = 0 , i = 0,1,2 , . M 1
[ d E i ( x ) dx ] 2 Q i 2 E i 2 ( x ) + 1 2 k 0 2 α i E i 4 ( x ) = C i
C i = [ d E i ( x ) dx ] 2 | x = x i Q i 2 E i 2 ( x i ) + 1 2 k 0 2 α i E i 4 ( x i )
d E i ( x ) dx = ± C i + Q i 2 E i 2 ( x ) 1 2 k 0 2 α i E i 4 ( x )
E i ( 0 ) E i ( x x i ) du ( a i 2 + u 2 ) ( b i 2 u 2 ) = ± ( 1 2 k 0 2 α i ) 1 2 0 x x i dx , x i x x i + 1
a i 2 = q i 2 Q i 2 k 0 2 α i
b i 2 = q i 2 Q i 2 k 0 2 α i
q i = ( Q i 4 + 2 C i k 0 2 α i ) 1 4
E i ( x ) = b i cn { q i [ ( x x i ) + x oci ] m i } , x i x x i + 1
m i = q i 2 + Q i 2 2 q i 2
E i ( x ) = A i cn [ q i ( x x i ) + B i m i ] cn [ B i m i ]
= A i cn [ q i ( x x i ) | m i ] { 1 fn [ q i ( x x i ) | m i ] fn [ B i | m i ] } 1 m i s n 2 [ q i ( x x i ) | m i ] s n 2 [ B i | m i ]
fn [ B i m i ] = sn [ B i m i ] dn [ B i m i ] cn [ B i m i ]
cn B i m i = A i b i
q 0 q 1 = fn B 1 m 1 fn B 0 m 0
q i 1 q i = fn B i m i fn q i 1 d i 1 + B i 1 m i 1 , 2 i M 1
fn [ q i d i m i ] = sn [ q i d i m i ] dn [ q i d i m i ] cn [ q i d i m i ]
fn [ q i d i + B i | m i ] = sn [ q i d i + B i | m i ] dn [ q i d i + B i | m i ] cn [ q i d i + B i | m i ]
= fn [ q i d i m i ] { d n 2 [ B i m i ] m i s n 2 [ B i m i ] c n 2 [ q i d i m i ] } { 1 fn [ q i d i m i ] fn [ B i m i ] } { 1 m i s n 2 [ q i d i m i ] s n 2 [ B i m i ] }
+ fn [ B i m i ] { d n 2 [ q i d i m i ] m i s n 2 [ q i d i m i ] c n 2 [ B i m i ] } { 1 fn [ q i d i m i ] fn [ B i m i ] } { 1 m i s n 2 [ q i d i m i ] s n 2 [ B i m i ] }
C i = A i 2 { [ q i fn ( q i d i + B i m i ) ] 2 Q i 2 + 1 2 k 0 2 α i A i 2 }
E i ( x ) = b i ˜ cn { q i ˜ [ ( x x i ) + x oci ] m i ˜ } , x i x x i + 1
a ˜ i 2 = q ˜ i 2 Q i 2 k 0 2 α i
b ˜ i 2 = q ˜ i 2 + Q i 2 k 0 2 α i
q i ˜ = i ( Q i 4 + 2 C i k 0 2 α i ) 1 4 = i q i
m i ˜ = q ˜ i 2 + Q i 2 2 q ˜ i 2
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = k 0 2 α 0 A 1 2 2 Q 0 2
, in the nonlinear substrate and nonlinear cladding
C i > 0 , q i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = 0
, in the linear substrate and linear cladding
C R > 0 , q R 2 = ( Q R 4 + 2 C R k 0 2 α R ) 1 4 = 0 , R { 1,3,5 , M 2 }
, in the arbitrary nonlinear guiding film
C i > 0 , q i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i | 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = 0
, in the linear substrate and linear cladding
C i > 0 , q i 2 = ( Q i 4 + 2 C i k 0 2 α i ) 1 4 , m i = q i 2 + Q i 2 2 q i 2
, i = 1,3 , M 2 , in the nonline ar guiding films
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = k 0 2 α 0 A 1 2 2 Q 0 2
, in the nonlinear substrate and nonlinear cladding
C i > 0 , q i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
tan ( Q i 1 d i 1 ) = Q i 1 [ q i tan ( B i ) + Q 0 tanh ( B 0 ) ] Q i 1 2 q i q 0 tan ( B i ) tanh ( B 0 ) , n e < n i 1
tanh ( Q i 1 d i 1 ) = Q i 1 [ q i tan ( B i ) + q 0 tanh ( B 0 ) ] Q i 1 2 + q i q 0 tan ( B i ) tanh ( B 0 ) , n e > n i 1
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = 0
, in the linear substrate and linear cladding
C R > 0 , q R 2 = ( Q R 4 + 2 C R k 0 2 α R ) 1 4 , R { 1,3,5 , , M 2 }
, in the arbitrary nonlinear guiding film
C i > 0 , q i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 1,3 , M 2 , in the guiding films , for n e < n i
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
q R + 1 b R + 2 cos ( B R + 2 ) b R + 2 q R + 2 sin ( B R + 2 ) 2 q R + 1 exp ( q R + 1 d R + 1 )
= { q R A R [ 1 - m R [ 1 - ( A R b R ) 2 ] ] sn [ q R d R ] dn [ q R d R ] q R A R m R [ 1 ( A R b R ) 2 ] sn [ q R d R ] c n 2 [ q R d R ] dn [ q R d R ] + ( A R q R 1 ) cn [ q R d R ] d n 2 [ q R d R ] ( A R q R 1 ) ( A R b R ) 2 m R s n 2 [ q R d R ] cn [ q R d R ] } 2 { 1 - m R [ 1 ( A R b R ) 2 s n 2 [ q R d R ] ] } 2
+ { A R cn [ q R d R m R ] ( A R q R ) q R sn [ q R d R | m R ] dn [ q R d R | m R ] } 2 { 1 m R [ 1 ( A R b R ) 2 ] s n 2 [ q R d R ] }
C 0 = C M 1 = 0 , q 0 2 = Q 0 2 , q M 1 2 = Q M 1 2 , m 0 = m M 1 = 1 , c n 2 [ q 0 x c 0 ] = 0
, in the linear substrate and linear cladding
C i > 0 , q i 2 = ( Q i 4 + 2 C i k 0 2 α i ) 1 4 , m i = q i 2 + Q i 2 2 q i 2
, i = 1,3 , M 2 , in the nonlinear guiding films
C i < 0 , q ˜ i 2 = Q i 2 , m i = 0 , i = 2,4 , M 3 , in the interaction layers
q R + 1 b R + 1 cos ( B R + 2 ) b R + 2 q R + 2 sin ( B R + 2 ) 2 q R + 1 exp ( q R + 1 d R + 1 )
= { q R A R [ 1 m R [ 1 ( A R b R ) 2 ] ] sn [ q R d R ] dn [ q R d R ] q R A R m R [ 1 ( A R b R ) 2 ] sn [ q R d R ] cn 2 [ q R D R ] dn [ q R d R ] + ( A R q R 1 ) cn [ q R d R ] dn 2 [ q R d R ] ( A R q R 1 ) ( A R b R ) 2 m R sn 2 [ q R d R ] cn [ q R d R ] } 2 2 { 1 m R [ 1 ( A R b R ) 2 ] sn 2 [ q R d R ] } 2 + { A R cn [ q R d R m R ] dn [ q R d R m R ] } 2 { 1 m R [ 1 ( A R b R ) 2 ] sn 2 [ q R d R ] }

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