Abstract

Much of nonlinear optics relates to the coupling of waves and changes in frequency components in the weakly nonlinear regime. The total field of a nonlinear optical waveguide can be expanded in terms of the modal fields of the linear waveguide, with the nonlinearity acting to couple power between the modes. For lossless systems there are at least two constants of the motion, one always being the conserved total power. The second constant has been constructed in various ways in specific problems and has sometimes been identified as a Hamiltonian. We show that a second constant can always be constructed by deriving a general formula for it in terms of the electromagnetic-field variables. Further, the second constant can then be used to write the coupled amplitude equations in Hamiltonian form. Specific examples are given.

© 1998 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  2. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1964).
  3. S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).
  4. R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
    [Crossref] [PubMed]
  5. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [Crossref]
  6. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [Crossref]
  7. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [Crossref]
  8. S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
    [Crossref]
  9. E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
    [Crossref]
  10. S. J. Garth and C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990);S. J. Garth and C. Pask, “Nonlinear effects in elliptical-core few-mode optical fibers,” J. Opt. Soc. Am. B 9, 243–250 (1992).
    [Crossref]
  11. Y. Chen, “Four-wave mixing in optical fibers: exact solution,” J. Opt. Soc. Am. B 6, 1986–1993 (1989).
    [Crossref]
  12. G. Capellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [Crossref]
  13. S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics: instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1992), pp. 489–534.
  14. A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
    [Crossref] [PubMed]
  15. A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
    [Crossref]
  16. D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” J. Lightwave Technol. 9, 1074–1082 (1991).
    [Crossref]
  17. Y. Chen, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
    [Crossref]
  18. W. Samir, S. J. Garth, and C. Pask, “Theory of fused-tapered nonlinear optical fiber couplers,” Appl. Opt. 32, 4513–4516 (1993).
    [Crossref] [PubMed]
  19. E. Caglioti, S. Trillo, S. Wabnitz, and G. I. Stegeman, “Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation,” J. Opt. Soc. Am. B 5, 472–482 (1988).
    [Crossref]
  20. W. Samir, S. J. Garth, and C. Pask, “Interplay of grating and nonlinearity in mode coupling,” J. Opt. Soc. Am. B 11, 64–71 (1994).
    [Crossref]
  21. S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibres,” Opt. Quantum Electron. 22, 37–53 (1990).
    [Crossref]
  22. Taking the time average implies that the higher-order harmonic terms vanish. This is equivalent to expanding both sides of Eq. (20) [using Eqs. (19)] and keeping the terms with the appropriate frequency components on each side.
  23. H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.
  24. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  25. D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).
  26. J. E. Sipe and G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
    [Crossref]
  27. B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
    [Crossref]
  28. B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
    [Crossref]
  29. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  30. R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
    [Crossref]
  31. W. Samir, “Nonlinear modal interactions in optical waveguides and devices,” Ph.D. dissertation (University of New South Wales, Canberra ACT, Australia, 1993), Subsection 2.4 and Appendix B.
  32. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 8.
  33. S. Trillo, S. Wabnitz, R. Chisari, and G. Capellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
    [Crossref] [PubMed]
  34. A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
    [Crossref]
  35. There is scope for confusion when comparing the formulas of this section with what appears in the literature. In order to be consistent with the notation developed in this paper, the even and odd supermodes of the twin-core nonlinear directional coupler have been labeled here as 1 and 2, respectively. Many authors working in this area have labelled them + and -, however, using the subscripts 1 and 2 for the modes of each individual core. The relationships between the two different descriptions are discussed briefly in Ref. 9.
  36. S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
    [Crossref]
  37. B. Crosignani and P. Di Porto, “Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers,” Opt. Acta 32, 1251–1258 (1985).
  38. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  39. J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
    [Crossref]
  40. B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
    [Crossref] [PubMed]
  41. J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).
  42. H. A. Haus and H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,” J. Opt. Soc. Am. A 66, 320–327 (1976).
    [Crossref]
  43. P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
    [Crossref]
  44. P. S. Pershan, “Nonlinear optics,” in Progress in Optics Vol. 5, E. Wolf, ed. (North-Holland, Amsterdam, 1966).
  45. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), Sect. 1.5.
  46. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sect. 10.
  47. M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single pulse equation,” Phys. Rev. A 35, 2965–2975 (1987).
    [Crossref] [PubMed]
  48. C. M. de Sterke and J. E. Sipe, “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
    [Crossref] [PubMed]
  49. F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
    [Crossref]

1994 (1)

1993 (1)

1992 (2)

S. Trillo, S. Wabnitz, R. Chisari, and G. Capellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[Crossref] [PubMed]

B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
[Crossref]

1991 (6)

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
[Crossref] [PubMed]

G. Capellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
[Crossref]

A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
[Crossref]

D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” J. Lightwave Technol. 9, 1074–1082 (1991).
[Crossref]

Y. Chen, “Mismatched nonlinear couplers with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 986–992 (1991).
[Crossref]

1990 (3)

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibres,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

S. J. Garth and C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990);S. J. Garth and C. Pask, “Nonlinear effects in elliptical-core few-mode optical fibers,” J. Opt. Soc. Am. B 9, 243–250 (1992).
[Crossref]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[Crossref]

1989 (3)

C. M. de Sterke and J. E. Sipe, “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref] [PubMed]

Y. Chen, “Four-wave mixing in optical fibers: exact solution,” J. Opt. Soc. Am. B 6, 1986–1993 (1989).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
[Crossref] [PubMed]

1988 (1)

1987 (2)

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single pulse equation,” Phys. Rev. A 35, 2965–2975 (1987).
[Crossref] [PubMed]

1986 (3)

A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[Crossref]

1985 (2)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

B. Crosignani and P. Di Porto, “Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers,” Opt. Acta 32, 1251–1258 (1985).

1982 (3)

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[Crossref] [PubMed]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[Crossref]

1979 (2)

J. E. Sipe and G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
[Crossref]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[Crossref]

1976 (1)

H. A. Haus and H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,” J. Opt. Soc. Am. A 66, 320–327 (1976).
[Crossref]

1975 (1)

F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
[Crossref]

1966 (1)

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

1963 (1)

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Ankiewicz, A.

A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Ashkin, A.

Assanto, G.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Betts, R. A.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1964).

Botineau, J.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), Sect. 1.5.

Brown, J.

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

Caglioti, E.

E. Caglioti, S. Trillo, S. Wabnitz, and G. I. Stegeman, “Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation,” J. Opt. Soc. Am. B 5, 472–482 (1988).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

Capellini, G.

Chen, Y.

Chisari, R.

Chu, P. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Crosignani, B.

B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
[Crossref]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[Crossref]

B. Crosignani and P. Di Porto, “Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers,” Opt. Acta 32, 1251–1258 (1985).

Daino, B.

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref] [PubMed]

Di Porto, P.

B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
[Crossref]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[Crossref]

B. Crosignani and P. Di Porto, “Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers,” Opt. Acta 32, 1251–1258 (1985).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Garth, S. J.

W. Samir, S. J. Garth, and C. Pask, “Interplay of grating and nonlinearity in mode coupling,” J. Opt. Soc. Am. B 11, 64–71 (1994).
[Crossref]

W. Samir, S. J. Garth, and C. Pask, “Theory of fused-tapered nonlinear optical fiber couplers,” Appl. Opt. 32, 4513–4516 (1993).
[Crossref] [PubMed]

S. J. Garth and C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990);S. J. Garth and C. Pask, “Nonlinear effects in elliptical-core few-mode optical fibers,” J. Opt. Soc. Am. B 9, 243–250 (1992).
[Crossref]

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibres,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

Glushko, B.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
[Crossref] [PubMed]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 8.

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Haus, H. A.

H. A. Haus and H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,” J. Opt. Soc. Am. A 66, 320–327 (1976).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[Crossref]

Karapetyan, A.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
[Crossref] [PubMed]

Khokhlov, R. V.

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Kogelnik, H.

H. A. Haus and H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,” J. Opt. Soc. Am. A 66, 320–327 (1976).
[Crossref]

H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

Kryzhanovsky, B.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
[Crossref] [PubMed]

Lam, J. F.

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sect. 10.

Lee, D. L.

D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sect. 10.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[Crossref]

Mitchell, D. J.

Pask, C.

W. Samir, S. J. Garth, and C. Pask, “Interplay of grating and nonlinearity in mode coupling,” J. Opt. Soc. Am. B 11, 64–71 (1994).
[Crossref]

W. Samir, S. J. Garth, and C. Pask, “Theory of fused-tapered nonlinear optical fiber couplers,” Appl. Opt. 32, 4513–4516 (1993).
[Crossref] [PubMed]

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibres,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

S. J. Garth and C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990);S. J. Garth and C. Pask, “Nonlinear effects in elliptical-core few-mode optical fibers,” J. Opt. Soc. Am. B 9, 243–250 (1992).
[Crossref]

Pershan, P. S.

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[Crossref]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

P. S. Pershan, “Nonlinear optics,” in Progress in Optics Vol. 5, E. Wolf, ed. (North-Holland, Amsterdam, 1966).

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Poladian, L.

Robinson, F. N. H.

F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
[Crossref]

Rowland, D. R.

A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
[Crossref]

D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” J. Lightwave Technol. 9, 1074–1082 (1991).
[Crossref]

Samir, W.

Seaton, C. T.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref] [PubMed]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single pulse equation,” Phys. Rev. A 35, 2965–2975 (1987).
[Crossref] [PubMed]

J. E. Sipe and G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
[Crossref]

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
[Crossref] [PubMed]

A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[Crossref]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Stegeman, G. I.

E. Caglioti, S. Trillo, S. Wabnitz, and G. I. Stegeman, “Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation,” J. Opt. Soc. Am. B 5, 472–482 (1988).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

J. E. Sipe and G. I. Stegeman, “Comparison of normal mode and total field analysis techniques in planar integrated optics,” J. Opt. Soc. Am. 69, 1676–1683 (1979).
[Crossref]

Stolen, R. H.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[Crossref] [PubMed]

Tjugiarto, T.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single pulse equation,” Phys. Rev. A 35, 2965–2975 (1987).
[Crossref] [PubMed]

Trillo, S.

S. Trillo, S. Wabnitz, R. Chisari, and G. Capellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[Crossref] [PubMed]

G. Capellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, and G. I. Stegeman, “Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation,” J. Opt. Soc. Am. B 5, 472–482 (1988).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics: instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1992), pp. 489–534.

Wabnitz, S.

S. Trillo, S. Wabnitz, R. Chisari, and G. Capellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992).
[Crossref] [PubMed]

E. Caglioti, S. Trillo, S. Wabnitz, and G. I. Stegeman, “Limitations to all-optical switching using nonlinear couplers in the presence of linear and nonlinear absorption and saturation,” J. Opt. Soc. Am. B 5, 472–482 (1988).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics: instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1992), pp. 489–534.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Xue, Y. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Yariv, A.

B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
[Crossref]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[Crossref]

Zheng, X.-H.

A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[Crossref]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, “Power-dependent switching in a coherent nonlinear directional coupler in the presence of saturation,” Appl. Phys. Lett. 51, 293–295 (1987).
[Crossref]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1986).
[Crossref]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[Crossref]

IEEE J. Quantum Electron. (3)

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[Crossref]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[Crossref]

J. Appl. Phys. (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

J. Lightwave Technol. (3)

S. J. Garth and C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” J. Lightwave Technol. 8, 129–137 (1990);S. J. Garth and C. Pask, “Nonlinear effects in elliptical-core few-mode optical fibers,” J. Opt. Soc. Am. B 9, 243–250 (1992).
[Crossref]

A. Ankiewicz, A. W. Snyder, and X.-H. Zheng, “Coupling between parallel optical fiber cores: critical examination,” J. Lightwave Technol. LT-4, 1317–1323 (1986).
[Crossref]

D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” J. Lightwave Technol. 9, 1074–1082 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

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

H. A. Haus and H. Kogelnik, “Electromagnetic momentum and momentum flow in dielectric waveguides,” J. Opt. Soc. Am. A 66, 320–327 (1976).
[Crossref]

J. Opt. Soc. Am. B (6)

Opt. Acta (1)

B. Crosignani and P. Di Porto, “Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers,” Opt. Acta 32, 1251–1258 (1985).

Opt. Commun. (2)

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[Crossref]

B. Crosignani, P. Di Porto, and A. Yariv, “Slowly-varying approximation and coupled-mode equations in guiding structures,” Opt. Commun. 91, 341–342 (1992).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibres,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

Phys. Rep. (1)

F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16, 313–354 (1975).
[Crossref]

Phys. Rev. (2)

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[Crossref]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Phys. Rev. A (3)

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material,” Phys. Rev. A 44, 6036–6042 (1991).
[Crossref] [PubMed]

M. V. Tratnik and J. E. Sipe, “Nonlinear polarization dynamics. I. The single pulse equation,” Phys. Rev. A 35, 2965–2975 (1987).
[Crossref] [PubMed]

C. M. de Sterke and J. E. Sipe, “Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Proc. IEEE (1)

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

Other (14)

P. S. Pershan, “Nonlinear optics,” in Progress in Optics Vol. 5, E. Wolf, ed. (North-Holland, Amsterdam, 1966).

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), Sect. 1.5.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sect. 10.

There is scope for confusion when comparing the formulas of this section with what appears in the literature. In order to be consistent with the notation developed in this paper, the even and odd supermodes of the twin-core nonlinear directional coupler have been labeled here as 1 and 2, respectively. Many authors working in this area have labelled them + and -, however, using the subscripts 1 and 2 for the modes of each individual core. The relationships between the two different descriptions are discussed briefly in Ref. 9.

W. Samir, “Nonlinear modal interactions in optical waveguides and devices,” Ph.D. dissertation (University of New South Wales, Canberra ACT, Australia, 1993), Subsection 2.4 and Appendix B.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), Chap. 8.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1964).

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics: instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1992), pp. 489–534.

Taking the time average implies that the higher-order harmonic terms vanish. This is equivalent to expanding both sides of Eq. (20) [using Eqs. (19)] and keeping the terms with the appropriate frequency components on each side.

H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Physics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).

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Figures (1)

Fig. 1
Fig. 1

Overview of the analysis.

Equations (71)

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Γ=m=1Lk=1NmβmkΠmk+1n+1 APtNLEt*+PtNL*EtdA,
dΠmkdz=-HBmk,
dBmkdz=HΠmk,
×E=-Bt,
×H=Dt,
D=0,
B=0,
B=μ0H,
D=E+PNL,
Et=m 12 [am(z)exp(iβmz)+bm(z)exp(-iβmz)]e¯t,m(x, y),
Ht=m 12 [am(z)exp(iβmz)-bm(z)exp(-iβmz)]h¯t,m(x, y),
z A(Et×H¯t*+E¯t*×Ht)zˆdA
=iωAE¯*PNLdA,
A(e¯t,m×h¯t,k*)zˆdA=A(e¯t,m*×h¯t,k)zˆdA=2skδm,k,
dakdz=iω2sk Ae¯k*PNLdA exp(-iβkz).
ak* dakdz=-ωskβk A 12 ak*(PtNLe¯t,mk*)×dA ddz [exp(-iβkz)].
Et,k*=12 ak*e¯t,k* exp(-iβkz).
dEt,k*dz=12 ak*e¯t,k* ddz [exp(-iβkz)]+12 e¯t,k* exp(-iβkz) dak*dz.
12 ak*(PtNLe¯t,k*) ddz [exp(-iβkz)]
=PtNL dEt,k*dz-12 (PtNLe¯t,k*)exp(-iβkz) dak*dz.
βkω skak* dakdz=-APtNL dEt,k*dzdA+dak*dz×A 12 (PtNLe¯t,k*)exp(-iβkz)dA.
βkω skak* dakdz=-APtNL dEt,m*dzdA-iω sk dak*dz dakdz.
βkω sk d|ak|2dz=-APtNL dEt,k*dz+PtNL* dEt,kdzdA.
ddz k=1N βkPkω=-APtNL dEt*dz+PtNL*dEtdzdA.
m=1Lk=1Nm βmkωm dPmkdz
=-m=1LAPmNL dEtm*dz+PmNL* dEtmdzdA
=-APNL dEt*dz+PNL* dEtdzdA,
P(3)=0χ(3)EEE,
P(3)=P(3)+P(3)*
E=E+E*.
ddz (P(3)E)=ddz (E0χ(3)EEE),
dP(3)dz E=34 ddz (P(3)E).
dP(3)dz E*+dP(3)*dz E=34 ddz P(3)E*+P(3)*E.
k=1N βkPkω+14 APt(3)Et*+Pt(3)*EtdA=Γ(3),
k=1N βkPkω+1n+1 APt(n)Et*+Pt(n)*EtdA
=Γ(n).
PNL=P(3)+P(5)+ .
Γsat=k=1N βkPkω+n 1n+1 APt(n)Et*+Pt(n)*EtdA,n=3, 5,  .
β1P1ω+β2P22ω+13 APt(2)Et*+Pt(2)*EtdA
=Γ(2),
skak=Pk exp(iϕk),
Pk=sk|ak|2.
sk d|ak|2dz=iωak*2 APtNLe¯t,k*dA exp(-iβkz)-ak2 APtNL*e¯t,kdA exp(iβkz).
dPkdz=iωA[PtNLEt,k*-PtNL*Et,k]dA.
dϕkdz=ω2Pk A[PtNLEt,k*+PtNL*Et,k]dA.
Bk=βkz+ϕk,
dBkdz=βk+ω2Pk A[PtNLEt,k*+PtNL*Et,k]dA.
d(Pk/ω)dz=-HBk,
dBkdz=H(Pk/ω),
h¯t,k*n00czˆ×e¯t,k*,
sk12 n00cAe¯t,k2dA,
E(x, y, z)=12 P1(z)s1 e¯1(x, y)exp(B1)+12 P2(z)s2 e¯2(x, y)exp(B2),
P(3)(ω)=0χ(3)[2(EE*)E+(EE)E*],
H=Γ=β1Π1+β2Π2+ω4 [Q11Π12+Q22Π22+4Q12Π1Π2+2Q12Π1Π2 cos 2(B1-B2)],
Qij=3ω0χ(3)8sisj Ae¯i2e¯j2dA,i, j=1, 2,
Ae¯i3e¯jdA=0,ij,
Γs=S2+Q2(β1-β2) S12,
S1=2P1P2 cos(B1-B2),
S2=P1-P2.
ωH=(β1-β2)2 Γs+Q4 P2+(β1+β2)2 P,
dΠxdz=-dΠydz=ωRΠxΠy sin 2θ,
dBxdz=βx+3ωR2 Πx+ωR2 Πy(2+cos 2θ),
dBydz=βy+3ωR2 Πy+ωR2 Πx(2+cos 2θ),
H=βxΠx+βyΠy+3ωR4 (Πx2+Πy2)+ωR2 ΠxΠy(2+cos 2θ).
E=12 a1(z)e¯1(x, y)exp(iβ1z-iω1t)+12 a2(z)e¯2(x, y)exp(iβ2z-iω2t)+c.c.
H=β1Π1+β2Π2+Rω1Π1ω2Π2 cos θ,
E=j=14 12 aj(z)e¯j(x, y)exp(iβjz-iωjt)+c.c.
H=j=14βjΠj+2c1ω1ω2ω3ω4Π1Π2Π3Π4×cos θ+c22 j=14k=14IjkωjωkΠjΠk,
Pn=-FEn*.
PnNL=-FNLEn*.
Γ=m,kβmkΠmk-AFNLdA.

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