Abstract

We propose a novel method for analyzing a multilayer optical waveguide structure with all nonlinear guiding films. This method can also be used to analyze a multibranch optical waveguide structure with all nonlinear guiding branches. The results show that agreement between theory and numerics is excellent.

© 2005 Optical Society of America

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References

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  1. G. I. Stegeman, C. T. Seaton, J Chilwell, and S. D. Smith, �??Nonlinear waves guided by thin films,�?? Apply. Phys. Lett. 44, 830 (1984).
    [CrossRef]
  2. A. D. Boardman and P. Egan, �??Optically nonlinear waves in thin films,�?? IEEE J. Quantum Electron. 22, 319 (1986).
    [CrossRef]
  3. A. D. Boardman and P. Egan, �??Nonlinear surface and guided polaritions of a general layered dielectric structure,�?? J. Phys. Colloq. C5, 291 (1984).
  4. N. N. Akhmediev, �??Novel class of nonlinear surface waves: Asymmetric modes in a symmetric layered structure,�?? Sov. Phys. �??JETP. 56, 299 (1982).
  5. F. Lederer, U. Langbein, and H. E. Ponath. �??Nonlinear waves guided by a dielectric slab,�?? Appl. Phys. B. 31, 187 (1983).
    [CrossRef]
  6. U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, �??Dispersion relations for nonlinear guided waves,�?? J. Mol. Struct. 115, 493 (1984).
    [CrossRef]
  7. D. Mihalache and H. Totia, �??S-polarized nonlinear surface and guided waves in an asymmetric layered structure,�?? Rev. Roumaine Phys. 29, 365 (1984).
  8. D. J. Robbins, �??TE modes in a slab waveguide bounded by nonlinear media,�?? Opt. Commun. 47, 309 (1983).
    [CrossRef]
  9. U. Langbein, F. Lederer, and H. E. Ponath, �??A new type of nonlinear slab-guided wave,�?? Opt. Commun. 46, 167 (1983).
    [CrossRef]
  10. F. Fedyanin and D. Mihalache, �??P-poiarized nonlinear surface polaritons in layered structures,�?? Z. Phys. B. 47, 167 (1982).
    [CrossRef]
  11. A. A. Maradudin, �??Nonlinear surface electromagnetic waves,�?? in Proc. 2nd Int. School Condensed Matter Phys., Varna, Bulgaria, Singapore: World Scientific (1983).
  12. D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, �??P-polarized nonlinear surface waves in symmetric layered structures.�?? Phys. Scripta. 29, 269 (1984).
    [CrossRef]
  13. A. E. Kaplan, �??Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,�?? Sov. Phys.-JETP. 45, 896 (1977).
  14. S. She and S. Zhang, �??Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,�?? Opt. Commmun. 161, 141 (1999).
    [CrossRef]
  15. Y. D. Wu, M. H. Chen, and H. J. Tasi, �??A General Method for Analyzing the Multilayer Optical Waveguide with Nonlinear Cladding and Substrate�??, SPIE Design, Fabrication, and Characterization of Photonic Dervice II, 4594. 323 (2001).
  16. Y. D. Wu and M. H. Chen, �??Analyzing multilayer optical waveguides with nonlinear cladding and substrates,�?? J. Opt. Soc. Am. B. 19, 1737, (2002).
    [CrossRef]
  17. Y. D. Wu and M. H. Chen, �??The fundamental theory of the symmetric three layer nonlinear optical waveguide structures and the numerical simulation,�?? J. Nat. Kao. Uni. of App. Sci., 32. 133 (2002).
  18. M. H. Chen, Y. D. Wu, and R. Z. Tasy, �??Analyses of antisymmetric modes of three-layer nonlinear optical waveguide,�?? J. Nat. Kao. Uni. of App. Sci., 34. 1 (2005)
  19. H. Murata, M. Izutsu, and T. Sueta, �??Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity,�?? J. Lightwave Technol. 16, 833 (1998).
    [CrossRef]
  20. Y. D. Wu, �??Analyzing multilayer optical waveguides with a localized arbitrary nonlinear guiding film,�?? IEEE J. Quantum Electron. 40, 529 (2004).
    [CrossRef]
  21. Y. D. Wu and D. H. Cai, �??Analytical and numerical analyses of TE-polarized waves in the planar optical waveguides with the nonlinear guiding film,�?? J. Eng. Tech. and Edu. 1 , 19 (2004).
  22. Yi-Fan Li and Keigo Iizuka, �??Unified Nonlinear Waveguide Dispersion Equations withtour Spurious Roots,�?? IEEE J. Quantum. Electron. 31, 791 (1995).
    [CrossRef]
  23. M. Cada, R. C. Gauthior, B. A. Paton, and J. Chrostowski, �??Nonlinear guided waves coupled nonlinearly in a planar GaAs/GaAlAs multiple-quantum-well structure,�?? Appl. Phys. Lett. 49, 755 (1986).
    [CrossRef]
  24. T. H. Wood, �??Multiple-quantum-well (MQW) waveguided modulator,�?? J. Lightwave Technol. 6, 743 (1988).
    [CrossRef]
  25. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, �??Third order nonlinear integrated moptics,�?? J. Lightwave Technol. 6, 953 (1988).
    [CrossRef]
  26. S. R. Cvetkovic and A. P. Zhao, �??Finite-element formalism for linear and nonlinear guided waves in multiple-quantum-well waveguides,�?? J. Opt. Soc. Amer. B. 10, 1401 (1993).
    [CrossRef]
  27. S. Selleri and M. Zobol, �??Stability analysis of nonlinear TE polarized waves in multiple-quantum-well waveguides,�?? IEEE J. Quantum Electron. 31, 1785 (1995).
    [CrossRef]
  28. C. J. Hamiltoin, J. H. Marsh, D. C. Hutchings, J. S. Aitchison, G. T. Kennedy, and W. Sibbett, �??Localized Kerr-type nonlinearities in GaAs/AlGaAs multiple quantum well structure at 1.55µm,�?? Appl. Phys. Lett. 68, 3078 (1996).
    [CrossRef]
  29. C. Rigo, L. Gastaldi, D. Campi, L. Faustini, C. Coriasso, C. Cacciatore and D. Sholdani, �??Multiple quantum well compressive strained heterostructures for low driving power all-optical waveguide switches,�?? J. Crystal. Growth. 188, 317 (1998).
    [CrossRef]
  30. Y. D. Wu, M. H. Chen, and C. H. Chu, �??All-optical logic device using bent nonlinear tapered Y-junction waveguide structure,�?? Fiber and Integrated Optics. 20, 517 (2001).
  31. Y. D. Wu, �??Nonlinear all-optical switching device by using the spatial soliton collision,�?? Fiber and Integrated Optics. 23, 387 (2004).
    [CrossRef]
  32. Y. D. Wu, �??New all-optical wavelength auto-router based on spatial solitons,�?? Opt. Express. 12, 4172 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4172">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4172</a>
    [CrossRef] [PubMed]
  33. Y. D. Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, �??Nonlinear all-optical phase and power-controlled switch by using the spatial solitons interaction,�?? SPIE. Bellingham. WA. 5646, 334 (2005).
  34. Y. D. Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, �??A new all-optical phase-controlled routing switch,�?? SPIE. Bellingham. WA. 5646, 345 (2005).
  35. Y. D. Wu, �??1�?N all-optical switching device by using the phase modulation of spatial solitons,�?? Applied. Optics. 44,4144(2005).
    [CrossRef] [PubMed]
  36. Y. D. Wu, �??All-optical logic gates by using multibranch waveguide structure with localized optical nonlinearity,�?? IEEE J. Sel. Top. Quantum. Electron. 11, 307 (2005).
    [CrossRef]
  37. T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, �??All-optical logic gates containing a two-mode nonlinear waveguide,�?? IEEE J. Quantum. Electron. 38, 37 (2002).
    [CrossRef]
  38. F. Garzia, and M. Bertolotti, �??All-optical security coded key,�?? Opt. Quantum. Electron. 33, 527, (2001).
    [CrossRef]
  39. Y. H. Pramono, and Endarko, �??Nonlinear waveguides for optical logic and computation,�?? J. Nonlinear Opt. Phys. Mater. 10, 209 (2001).
    [CrossRef]
  40. W. K. Burns, and A. F. Milton, �??Mode conversion in planar dielectric separating waveguide,�?? IEEE J. Quantum. Electron. 11, 32 (1975).
    [CrossRef]
  41. D. Marcuse, �??Radiation losses of tapered dielectric slab wave-guides,�?? Bell Syst; Tech. J., 49, 273, (1970).
    [PubMed]
  42. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and D. Smith, �??Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,�?? IEEE J. Quantum. Electron. 21, 774, (1985).
    [CrossRef]
  43. C. T. Seaton, X. Mai, G. I. Stegeman, N. G. Winful, �??Nonlinear guided wave applications,�?? Opt. Eng., 24, 593 (1985).
  44. H. Vach, G. I. Stegeman, C. T. Seaton, and I. C. Khoo, �??Experimental observation of nonlinear guided waves,�?? Opt. Lett. 9, 238 (1984).
    [CrossRef] [PubMed]
  45. H. F. Chou, C. F. Lin, and G. C. Wang, �??An Interative Finite Difference Beam Propagation Method for Modeling Second-Order Nonlinear Effects in Optical Waveguides,�?? J. Lightwave. Technol. 16, 1686 (1998).
    [CrossRef]

Appl. Phys. B.

F. Lederer, U. Langbein, and H. E. Ponath. �??Nonlinear waves guided by a dielectric slab,�?? Appl. Phys. B. 31, 187 (1983).
[CrossRef]

Appl. Phys. Lett.

M. Cada, R. C. Gauthior, B. A. Paton, and J. Chrostowski, �??Nonlinear guided waves coupled nonlinearly in a planar GaAs/GaAlAs multiple-quantum-well structure,�?? Appl. Phys. Lett. 49, 755 (1986).
[CrossRef]

C. J. Hamiltoin, J. H. Marsh, D. C. Hutchings, J. S. Aitchison, G. T. Kennedy, and W. Sibbett, �??Localized Kerr-type nonlinearities in GaAs/AlGaAs multiple quantum well structure at 1.55µm,�?? Appl. Phys. Lett. 68, 3078 (1996).
[CrossRef]

Applied. Optics.

Y. D. Wu, �??1�?N all-optical switching device by using the phase modulation of spatial solitons,�?? Applied. Optics. 44,4144(2005).
[CrossRef] [PubMed]

Apply. Phys. Lett.

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

Bell Syst; Tech. J.

D. Marcuse, �??Radiation losses of tapered dielectric slab wave-guides,�?? Bell Syst; Tech. J., 49, 273, (1970).
[PubMed]

Fiber and Integrated Optics.

Y. D. Wu, M. H. Chen, and C. H. Chu, �??All-optical logic device using bent nonlinear tapered Y-junction waveguide structure,�?? Fiber and Integrated Optics. 20, 517 (2001).

Y. D. Wu, �??Nonlinear all-optical switching device by using the spatial soliton collision,�?? Fiber and Integrated Optics. 23, 387 (2004).
[CrossRef]

IEEE J. Quantum Electron.

S. Selleri and M. Zobol, �??Stability analysis of nonlinear TE polarized waves in multiple-quantum-well waveguides,�?? IEEE J. Quantum Electron. 31, 1785 (1995).
[CrossRef]

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

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

IEEE J. Quantum. Electron.

Yi-Fan Li and Keigo Iizuka, �??Unified Nonlinear Waveguide Dispersion Equations withtour Spurious Roots,�?? IEEE J. Quantum. Electron. 31, 791 (1995).
[CrossRef]

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

T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, �??All-optical logic gates containing a two-mode nonlinear waveguide,�?? IEEE J. Quantum. Electron. 38, 37 (2002).
[CrossRef]

W. K. Burns, and A. F. Milton, �??Mode conversion in planar dielectric separating waveguide,�?? IEEE J. Quantum. Electron. 11, 32 (1975).
[CrossRef]

IEEE J. Sel. Top. Quantum. Electron.

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

J. Crystal. Growth.

C. Rigo, L. Gastaldi, D. Campi, L. Faustini, C. Coriasso, C. Cacciatore and D. Sholdani, �??Multiple quantum well compressive strained heterostructures for low driving power all-optical waveguide switches,�?? J. Crystal. Growth. 188, 317 (1998).
[CrossRef]

J. Eng. Tech. and Edu.

Y. D. Wu and D. H. Cai, �??Analytical and numerical analyses of TE-polarized waves in the planar optical waveguides with the nonlinear guiding film,�?? J. Eng. Tech. and Edu. 1 , 19 (2004).

J. Lightwave Technol.

H. Murata, M. Izutsu, and T. Sueta, �??Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity,�?? J. Lightwave Technol. 16, 833 (1998).
[CrossRef]

T. H. Wood, �??Multiple-quantum-well (MQW) waveguided modulator,�?? J. Lightwave Technol. 6, 743 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, �??Third order nonlinear integrated moptics,�?? J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

J. Lightwave. Technol.

H. F. Chou, C. F. Lin, and G. C. Wang, �??An Interative Finite Difference Beam Propagation Method for Modeling Second-Order Nonlinear Effects in Optical Waveguides,�?? J. Lightwave. Technol. 16, 1686 (1998).
[CrossRef]

J. Mol. Struct.

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, �??Dispersion relations for nonlinear guided waves,�?? J. Mol. Struct. 115, 493 (1984).
[CrossRef]

J. Nat. Kao. Uni. of App. Sci.

Y. D. Wu and M. H. Chen, �??The fundamental theory of the symmetric three layer nonlinear optical waveguide structures and the numerical simulation,�?? J. Nat. Kao. Uni. of App. Sci., 32. 133 (2002).

M. H. Chen, Y. D. Wu, and R. Z. Tasy, �??Analyses of antisymmetric modes of three-layer nonlinear optical waveguide,�?? J. Nat. Kao. Uni. of App. Sci., 34. 1 (2005)

J. Nonlinear Opt. Phys. Mater.

Y. H. Pramono, and Endarko, �??Nonlinear waveguides for optical logic and computation,�?? J. Nonlinear Opt. Phys. Mater. 10, 209 (2001).
[CrossRef]

J. Opt. Soc. Am. B.

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

J. Opt. Soc. Amer. B.

S. R. Cvetkovic and A. P. Zhao, �??Finite-element formalism for linear and nonlinear guided waves in multiple-quantum-well waveguides,�?? J. Opt. Soc. Amer. B. 10, 1401 (1993).
[CrossRef]

J. Phys. Colloq.

A. D. Boardman and P. Egan, �??Nonlinear surface and guided polaritions of a general layered dielectric structure,�?? J. Phys. Colloq. C5, 291 (1984).

Opt. Commmun.

S. She and S. Zhang, �??Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,�?? Opt. Commmun. 161, 141 (1999).
[CrossRef]

Opt. Commun.

D. J. Robbins, �??TE modes in a slab waveguide bounded by nonlinear media,�?? Opt. Commun. 47, 309 (1983).
[CrossRef]

U. Langbein, F. Lederer, and H. E. Ponath, �??A new type of nonlinear slab-guided wave,�?? Opt. Commun. 46, 167 (1983).
[CrossRef]

Opt. Eng.

C. T. Seaton, X. Mai, G. I. Stegeman, N. G. Winful, �??Nonlinear guided wave applications,�?? Opt. Eng., 24, 593 (1985).

Opt. Express.

Y. D. Wu, �??New all-optical wavelength auto-router based on spatial solitons,�?? Opt. Express. 12, 4172 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4172">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4172</a>
[CrossRef] [PubMed]

Opt. Lett.

Opt. Quantum. Electron.

F. Garzia, and M. Bertolotti, �??All-optical security coded key,�?? Opt. Quantum. Electron. 33, 527, (2001).
[CrossRef]

Phys. Scripta.

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, �??P-polarized nonlinear surface waves in symmetric layered structures.�?? Phys. Scripta. 29, 269 (1984).
[CrossRef]

Proc. SPIE

Y. D. Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, �??A new all-optical phase-controlled routing switch,�?? SPIE. Bellingham. WA. 5646, 345 (2005).

Rev. Roumaine Phys.

D. Mihalache and H. Totia, �??S-polarized nonlinear surface and guided waves in an asymmetric layered structure,�?? Rev. Roumaine Phys. 29, 365 (1984).

School Condensed Matter Phys 1983

A. A. Maradudin, �??Nonlinear surface electromagnetic waves,�?? in Proc. 2nd Int. School Condensed Matter Phys., Varna, Bulgaria, Singapore: World Scientific (1983).

Sov. Phys. JETP

N. N. Akhmediev, �??Novel class of nonlinear surface waves: Asymmetric modes in a symmetric layered structure,�?? Sov. Phys. �??JETP. 56, 299 (1982).

Sov. Phys.-JETP.

A. E. Kaplan, �??Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,�?? Sov. Phys.-JETP. 45, 896 (1977).

SPIE Design, Fabrication, and Char. of P

Y. D. Wu, M. H. Chen, and H. J. Tasi, �??A General Method for Analyzing the Multilayer Optical Waveguide with Nonlinear Cladding and Substrate�??, SPIE Design, Fabrication, and Characterization of Photonic Dervice II, 4594. 323 (2001).

SPIE.

Y. D. Wu, Y. F. Laio, M. H. Chen, and K. H. Chiang, �??Nonlinear all-optical phase and power-controlled switch by using the spatial solitons interaction,�?? SPIE. Bellingham. WA. 5646, 334 (2005).

Z. Phys. B.

F. Fedyanin and D. Mihalache, �??P-poiarized nonlinear surface polaritons in layered structures,�?? Z. Phys. B. 47, 167 (1982).
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

The structure of multilayer optical waveguides with all nonlinear guiding films.

Fig. 2.
Fig. 2.

Diagram of the computation steps (9)–(11)

Fig. 3.
Fig. 3.

(a)Dispersion curve of the three-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

Fig. 4.
Fig. 4.

(a)Dispersion curve of the five-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

Fig. 5.
Fig. 5.

(a)Dispersion curve of the seven-layer optical waveguide structure with the nonlinear central guiding film. (b)The electric field distributions with respect to points A-D as shown in (a).

Fig. 6.
Fig. 6.

The multibranch optical waveguide structure with all nonlinear guiding branches (N denoted the nonlinear medium).

Fig. 7.
Fig. 7.

The three-branch optical waveguide structure with all nonlinear guiding branches (N denoted the nonlinear medium).

Fig. 8.
Fig. 8.

For the low input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear guiding branches at positions (a)Z1 (df = 3μm , β = 1.5654), (b)Z2 (df = 6μm, β = 1.5654), (c)Z3 (df = 3μm , dI = 5μm , β = 1.5655), (d)Z4 (df = 3μm , dI = 7μm, β = 1.56545).

Fig. 9.
Fig. 9.

For the high input power density case, electric-field distributions of the three-branch optical waveguide structure with all nonlinear guiding branches at positions (a)Z1 (df = 3μm , β = 1.56925), (b)Z2 (df = 6μm, β = 1.56925), (c)Z3 (df = 3μm , dI = 5μm, β = 1.5690), (d)Z4 (df = 3μm, dI = 7μm, β = 1.5693).

Fig. 10.
Fig. 10.

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

Fig. 11.
Fig. 11.

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

Fig. 12.
Fig. 12.

The relevant curves shown in Figs. 6 and 8 on the same graph (a) at position Z1, (b) at position Z2, (c) at position Z3, (d) at position Z4. (┄:the predicted results;―: the numerical simulation results).

Fig. 13.
Fig. 13.

The relevant curves shown in Figs. 7 and 9 on the same graph (a) at position Z1, (b) at position Z2, (c) at position Z3, (d) at position Z4. (┄:the predicted results;―: the numerical simulation results).

Equations (49)

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2 E yi = n i 2 c 2 2 E yi t 2 , i = 1,2 , , m
E yi x z t = ε i ( x ) exp [ j ( ωt βk 0 z ) ]
n i 2 = n 0 i 2 + α i ε i ( x ) 2 , i = 2,4 , , m 1
ε 1 ( x ) = E s exp ( p 1 x ) in the substrate
ε i ( x ) = E I ( i 2 ) exp { p i [ x ( i 1 2 ) d ( i 3 2 ) w ] } + E I ( i 1 ) { exp p i [ x ( i 1 2 ) d ( i 1 2 ) w ] }
i = 3,5 , , m 2 in the interaction layers
ε i ( x ) = b i cn { A i [ x ( i 2 1 ) ( d + w ) + x 0 i ] l i }
i = 2,4 , , m 1 in the guiding film , for β < n i
ε i ( x ) = b i ¯ cn { A i ¯ [ x ( i 2 1 ) ( d + w ) + x ̅ 0 i ] l i ¯ }
i = 2,4 , , m 1 in the guiding film , for β > n i
ε m ( x ) = E c exp { p m [ x ( m 1 2 ) d ( m 3 2 ) w ] } in the cladding
p i = k 0 β 2 n i 2 ,
b i 2 = q i 4 + 2 α i k 0 2 K i q i 2 α i k 0 2 ,
A i = [ ( a i 2 + b i 2 ) ( α i k 0 2 2 ) ] 1 2 ,
l i = b i 2 ( a i 2 + b i 2 ) ,
b ̅ i 2 = Q i 4 + 2 α i k 0 2 K i + Q i 2 α i k 0 2 ,
A ̅ i = [ ( a ̅ i 2 + b ̅ i 2 ) ( α i k 0 2 2 ) ] 1 2 ,
l ̅ i = b ̅ i 2 ( a ̅ i 2 + b ̅ i 2 ) ,
[ cn ( A m 1 d ) Δ m 2 + + sn ( A m 1 d ) dn ( A m 1 d ) p m 2 Δ m 2 A m 1 ] [ 1 l m 1 ( 1 Δ m 2 + 2 b m 1 2 ) sn 2 ( A m 1 d ) ] A m 1 = { Δ m 2 + [ 1 l m 1 ( 1 Δ m 2 + 2 b m 1 2 ) sn ( A m 1 d ) dn ( A m 1 d ) ] Δ m 2 + l m 1 ( 1 Δ m 2 + 2 b m 1 2 ) sn ( A m 1 d ) cn 2 ( A m 1 d ) dn ( A m 1 d ) + p m 2 Δ m 2 A m 1 [ cn ( A m 1 d ) dn 2 ( A m 1 d ) l m 1 Δ m 2 + 2 b m 1 4 sn 2 ( A m 1 d ) cn ( A m 1 d ) ] } [ 1 l m 1 ( 1 Δ m 2 + 2 b m 1 4 ) sn 2 ( A m 1 d ) ] 2 p m ,
where Δ m 2 + = E I ( m 4 ) exp ( p m 2 w ) + E I ( m 3 ) ,
Δ m 2 = E I ( m 4 ) exp ( p m 2 w ) E I ( m 3 ) .
E c P 3 A 2 b 2 = { E s b 2 [ 1 l 2 ( 1 E s 2 b 2 2 ) ] sn ( A 2 d ) dn ( A 2 d ) l 2 E s b 2 ( 1 E s 2 b 2 2 ) sn ( A 2 d ) cn 2 ( A 2 d ) dn ( A 2 d ) E s p 1 b 2 A 2 [ cn ( A 2 d ) dn 2 ( A 2 d ) l 2 E s 2 b 2 2 sn 2 ( A 2 d ) cn ( A 2 d ) ] } [ E s E c cn ( A 2 d ) + E s p 1 E c A 2 sn ( A 2 d ) dn ( A 2 d ) ] 2 ,
for m = 5 ,
[ Δ 3 + cn ( A 4 d ) p 3 Δ 3 A 4 sn ( A 4 d ) dn ( A 4 d ) ] [ 1 l 4 ( 1 Δ 3 + 2 b 4 2 ) sn 2 ( A 4 d ) ] A 4 = { Δ 3 + [ 1 l 4 ( 1 Δ 3 + 2 b 4 2 ) ] sn ( A 4 d ) dn ( A 4 d ) Δ 3 + l 4 ( 1 Δ 3 + 2 b 4 2 ) sn ( A 4 d ) dn ( A 4 d ) cn 2 ( A 4 d ) } + p 3 Δ 3 A 4 cn ( A 4 d ) [ dn 2 ( A 4 d ) l 4 Δ 3 + 2 b 4 2 sn 2 ( A 4 d ) ] [ 1 l 4 ( 1 Δ 3 + 2 b 4 2 ) sn 2 ( A 4 d ) ] 2 p 5 ,
where Δ 3 + = E I 1 exp ( p 3 w ) + E I 2 ,
Δ 3 = E I 1 exp ( p 3 w ) + E I 2 ,
a i 2 = q i 4 + 2 α i k 0 2 K i + q i 2 α i k 0 2 ,
a ̅ i 2 = q i 4 + 2 α i k 0 2 K i Q i 2 α i k 0 2 ,
A i = [ ( a i 2 + b i 2 ) ( α i k 0 2 2 ) ] 1 2 ,
A ̅ i = [ ( a ̅ i 2 + b ̅ i 2 ) ( α i k 0 2 2 ) ] 1 2 ,
q i 2 = k 0 ( n i 2 β 2 ) , for β < n i ( n i = 2,4 , , m 1 ) ,
Q i 2 = k 0 ( β 2 n i 2 ) , for β > n i ( n i = 2,4 , , m 1 ) ,
E I 1 = 1 2 { [ E s cn ( A 2 d ) + p 1 E s A 2 sn ( A 2 d ) dn ( A 2 d ) ] [ 1 l 2 sn 2 ( A 2 d ) ( 1 E s 2 b 2 2 ) ] }
+ A 2 { E s sn ( A 2 d ) dn ( A 2 d ) { [ 1 l 2 ( 1 E s 2 b 2 2 ) ] l 2 ( 1 E s 2 b 2 2 ) cn 2 ( A 2 d ) } p 1 E s A 2 cn ( A 2 d ) [ dn 2 ( A 2 d ) l 2 E s 2 b 2 2 sn 2 ( A 2 d ) ] p 3 [ 1 l 2 sn 2 ( A 2 d ) ( 1 E s 2 b 2 2 ) ] 2 } ,
E Ii = 1 2 { [ Δ i 2 + cn ( A i + 1 d ) + p i 2 Δ i 2 A i + 1 ] sn ( A i + 1 ) dn ( A i + 1 d ) [ 1 l i + 1 ( 1 Δ i 2 + 2 b i + 1 2 ) sn 2 ( A i + 1 d ) ] + A i + 1 { Δ i 2 + [ 1 l i + 1 ( 1 Δ i 2 + 2 b i + 1 2 ) ] [ 1 l i + 1 cn 2 ( A i + 1 ) ] sn ( A i + 1 d ) dn ( A i + 1 d ) Δ i 2 + l i + 1 ( 1 Δ i 2 + 2 b i + 1 4 ) sn ( A i + 1 d ) cn 2 ( A i + 1 d ) dn ( A i + 1 d ) Δ i 2 p i 2 A i + 1 cn ( A i + 1 d ) [ dn 2 ( A i + 1 d ) l i + 1 Δ i 2 + 2 b i + 1 2 sn 2 ( A i + 1 d ) ] } [ 1 l i + 1 sn 2 ( A i + 1 d ) ( 1 Δ i 2 + 2 b i + 1 4 ) ] 2 p i } , i = 3,5 , m 2 ,
where Δ i 2 + = E Ii 2 + E Ii 1 exp ( p i 2 w ) ,
Δ i 2 = E Ii 2 E Ii 1 exp ( p i 2 w ) .
For i = 4,6 , m 3 ,
x 0 i = 1 A i cn 1 ( E I ( i 3 ) exp ( p ( i 1 ) w ) + E I ( i 2 ) b i ) ,
x ̅ 0 i = 1 A i ¯ cn 1 ( E I ( i 3 ) exp ( p ( i 1 ) w ) + E I ( i 2 ) b i ¯ ) ,
K i = k 0 2 E I ( i 1 ) 2 ( n i 2 n i + 1 2 + α i E I ( i 1 ) 2 2 ) + k 0 2 E Ii 2 exp ( 2 p ( i + 1 ) w ) [ n i 2 n i + 1 2 + α i E Ii 2 2 exp ( 2 p ( i + 1 ) w ) ]
+ 2 E I ( i 1 ) E Ii k 0 2 exp ( p ( i + 1 ) w ) [ n i 2 2 β 2 + n i + 1 2 + 3 2 α i E I ( i 1 ) E Ii exp ( p ( i + 1 ) w ) ]
+ 2 α i E I ( i 1 ) E Ii k 0 2 exp ( p ( i + 1 ) w ) [ E I ( i 1 ) 2 + E Ii 2 exp ( 2 p ( i + 1 ) w ) ] .
x 02 = 1 A 2 cn 1 ( E 0 b 2 ) ,
x 0 m 1 = 1 A m 1 cn 1 ( E 0 b m 1 ) d ,
x ̅ 02 = 1 A 2 ¯ cn 1 ( E 0 b ̅ 2 ) ,
x ̅ 0 m 1 = 1 A ̅ m 1 cn 1 ( E 0 b ̅ m 1 ) d ,
K 2 = k 0 2 E 0 2 ( n 2 2 n 1 2 + α 2 E 0 2 2 ) ,
K m 1 = k 0 2 E m 2 ( n m 1 2 n m 2 + α ( m 1 ) E m 2 2 ) .

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