Abstract

The equations for Four-Wave-Mixing in a photonic crystal waveguide are derived accurately in the hypotesis of negligible nonlinear absorption. The dispersive nature of slow-light enhancement, the impact of Bloch mode reshaping in the nonlinear overlap integrals and the tensor nature of the third order polarization are therefore taken into account. Numerical calculations reveal substantial differences from simpler models, which increase with decreasing group velocity. We predict that the gain for a 1.3 mm long, un-optimized GaInP waveguide will exceed 10 dB if the pump power exceeds 1 W.

© 2010 Optical Society of America

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References

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  1. . Slow Light: Science and Applications, J. B. Khurgin and R. S. Tucker, Eds., (CRC Press, Boca Raton, 2009).
  2. . M. Santagiustina, “Governing the speed of light: recent advances and future perspectives of slow and fast light in microwave-photonics”, in Proc. 2009 Intern. Top. Meet. on Microwave Photonics, (Valencia, Spain, 2009) Th3.1.
  3. . T. Baba, “Slow light in photonic crystals.” Nat. Phot. 2, 465–473 (2008).
    [CrossRef]
  4. . N. A. R. Bhat, and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals”. Phys. Rev. E 64, 056604 (2001).
    [CrossRef]
  5. . M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
    [CrossRef]
  6. . T. Kamalakis, and T. Sphicopoulos, “A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of Kerr-induced nonlinearities,” IEEE J. Quantum Electron. 43, 923-933 (2007).
    [CrossRef]
  7. . T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
    [CrossRef]
  8. . B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
    [CrossRef]
  9. . B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Optical signal processing on a silicon chip at 640Gb/s using slow-light,” Opt. Express 18, 7770-7781 (2010).
    [CrossRef] [PubMed]
  10. . S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
    [CrossRef]
  11. . C. Husko, S. Combrié, Q. Tran, F. Raineri, C. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorptionin III-V photonic crystal waveguides,” Opt. Express 17, 22442-22451 (2009).
    [CrossRef]
  12. . V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. 35, 1440-1442 (2010).
    [CrossRef] [PubMed]
  13. . T. Hasegawa, T. Nagashima, and N. Sugimoto, “Determination of nonlinear coefficient and group-velocity dispersion of bismuth-based high nonlinear optical fiber by four-wave mixing,” Opt. Commun. 281, 782-787 (2008).
    [CrossRef]
  14. . M. D. Pelusi, F. Luan, E. Magi, M. R. E. Lamont, D. J. Moss, B. J. Eggleton, J. S. Sanghera, L. B. Shaw, and I. D. Aggarwal, “High bit rate all-optical signal processing in a fiber photonic wire,” Opt. Express 16, 11506-11512 (2008).
    [CrossRef] [PubMed]
  15. . M. Ebnali-Heidari, C. Monat, C. Grillet, and M. K. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express 17, 18340-18353 (2009).
    [CrossRef] [PubMed]
  16. . http://ab-initio.mit.edu/photons/
  17. . N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. T. Quantum Electron. 16, 257-266 (2010).
    [CrossRef]
  18. . D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
    [CrossRef]
  19. . R. Boyd, Nonlinear Optics Chapt. 4 (Academic Press, San Diego, 2003).
  20. . P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742 (1979).
    [CrossRef]
  21. . K. Sakoda, Optical Properties of Photonic Crystals, Chapt. 2 (Springer, Berlin, 2005).
  22. . B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
    [CrossRef]
  23. . G. P. Agrawal, Nonlinear fiber optics, Chapt. 10 (Academic Press, San Diego, 2001).
  24. . D. C. Hutchings, and B. S. Wherrett, “Polarisation dichroism of nonlinear refraction in zinc-blende semiconductors,” Opt. Commun. 111, 507–512 (1994).
    [CrossRef]
  25. . J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227-6232 (2008).
    [CrossRef] [PubMed]

2010

2009

. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. K. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express 17, 18340-18353 (2009).
[CrossRef] [PubMed]

. C. Husko, S. Combrié, Q. Tran, F. Raineri, C. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorptionin III-V photonic crystal waveguides,” Opt. Express 17, 22442-22451 (2009).
[CrossRef]

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
[CrossRef]

2008

2007

. T. Kamalakis, and T. Sphicopoulos, “A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of Kerr-induced nonlinearities,” IEEE J. Quantum Electron. 43, 923-933 (2007).
[CrossRef]

. T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
[CrossRef]

2005

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

2003

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

2002

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

2001

. N. A. R. Bhat, and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals”. Phys. Rev. E 64, 056604 (2001).
[CrossRef]

1994

. D. C. Hutchings, and B. S. Wherrett, “Polarisation dichroism of nonlinear refraction in zinc-blende semiconductors,” Opt. Commun. 111, 507–512 (1994).
[CrossRef]

1979

Aggarwal, I. D.

Baba, T.

. T. Baba, “Slow light in photonic crystals.” Nat. Phot. 2, 465–473 (2008).
[CrossRef]

Bhat, N. A. R.

. N. A. R. Bhat, and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals”. Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Braüer, A.

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

Cestier, I.

Colman, P.

. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. G. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. 35, 1440-1442 (2010).
[CrossRef] [PubMed]

. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
[CrossRef]

Combrié, S.

Corcoran, B.

. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Optical signal processing on a silicon chip at 640Gb/s using slow-light,” Opt. Express 18, 7770-7781 (2010).
[CrossRef] [PubMed]

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

De Rossi, A.

Dunbar, L. A.

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

Ebnali-Heidari, M.

Eckhouse, V.

Eggleton, B. J.

Eisenstein, G.

Ferrini, R.

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

Fink, Y.

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Gomez-Iglesias, A.

Grillet, C.

Hasegawa, T.

. T. Hasegawa, T. Nagashima, and N. Sugimoto, “Determination of nonlinear coefficient and group-velocity dispersion of bismuth-based high nonlinear optical fiber by four-wave mixing,” Opt. Commun. 281, 782-787 (2008).
[CrossRef]

Houdre, R.

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

Husko, C.

. C. Husko, S. Combrié, Q. Tran, F. Raineri, C. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorptionin III-V photonic crystal waveguides,” Opt. Express 17, 22442-22451 (2009).
[CrossRef]

. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
[CrossRef]

Hutchings, D. C.

. D. C. Hutchings, and B. S. Wherrett, “Polarisation dichroism of nonlinear refraction in zinc-blende semiconductors,” Opt. Commun. 111, 507–512 (1994).
[CrossRef]

Ibanescu, M.

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Joannopoulos, J. D.

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Johnson, S. G.

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Kamalakis, T.

. T. Kamalakis, and T. Sphicopoulos, “A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of Kerr-induced nonlinearities,” IEEE J. Quantum Electron. 43, 923-933 (2007).
[CrossRef]

Krauss, T.

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

Krauss, T. F.

Lamont, M. R. E.

Li, J.

Lombardet, B.

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

Luan, F.

Magi, E.

McMillan, J. F.

. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. T. Quantum Electron. 16, 257-266 (2010).
[CrossRef]

Michaelis, D.

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

Monat, C.

Moravvej-Farshi, M. K.

Moss, D.

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

Moss, D. J.

Nagashima, T.

. T. Hasegawa, T. Nagashima, and N. Sugimoto, “Determination of nonlinear coefficient and group-velocity dispersion of bismuth-based high nonlinear optical fiber by four-wave mixing,” Opt. Commun. 281, 782-787 (2008).
[CrossRef]

O’Faolain, L.

Panoiu, N. C.

. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. T. Quantum Electron. 16, 257-266 (2010).
[CrossRef]

Pelusi, M.

Pelusi, M. D.

Peschel, U.

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

Raineri, F.

Sanghera, J. S.

Santagiustina, M.

Shaw, L. B.

Sipe, J. E.

. N. A. R. Bhat, and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals”. Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Soljacic, M.

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Someda, C. G.

Sphicopoulos, T.

. T. Kamalakis, and T. Sphicopoulos, “A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of Kerr-induced nonlinearities,” IEEE J. Quantum Electron. 43, 923-933 (2007).
[CrossRef]

Sugimoto, N.

. T. Hasegawa, T. Nagashima, and N. Sugimoto, “Determination of nonlinear coefficient and group-velocity dispersion of bismuth-based high nonlinear optical fiber by four-wave mixing,” Opt. Commun. 281, 782-787 (2008).
[CrossRef]

Tran, Q.

Vadalà, G.

Vy Tran, Q.

. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
[CrossRef]

Wächter, C.

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

Wherrett, B. S.

. D. C. Hutchings, and B. S. Wherrett, “Polarisation dichroism of nonlinear refraction in zinc-blende semiconductors,” Opt. Commun. 111, 507–512 (1994).
[CrossRef]

White, T.

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

White, T. P.

Wong, C.

Wong, C. W.

. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. T. Quantum Electron. 16, 257-266 (2010).
[CrossRef]

Yeh, P.

Appl. Phys. Lett.

. S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. 95, 221108 (2009).
[CrossRef]

IEEE J. Quantum Electron.

. T. Kamalakis, and T. Sphicopoulos, “A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of Kerr-induced nonlinearities,” IEEE J. Quantum Electron. 43, 923-933 (2007).
[CrossRef]

IEEE J. Sel. T. Quantum Electron.

. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. T. Quantum Electron. 16, 257-266 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D: Appl. Phys.

. T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666-2670 (2007).
[CrossRef]

Nat. Phot.

. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. J. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Phot. 3, 206-210 (2009).
[CrossRef]

. T. Baba, “Slow light in photonic crystals.” Nat. Phot. 2, 465–473 (2008).
[CrossRef]

Opt. Commun.

. T. Hasegawa, T. Nagashima, and N. Sugimoto, “Determination of nonlinear coefficient and group-velocity dispersion of bismuth-based high nonlinear optical fiber by four-wave mixing,” Opt. Commun. 281, 782-787 (2008).
[CrossRef]

. D. C. Hutchings, and B. S. Wherrett, “Polarisation dichroism of nonlinear refraction in zinc-blende semiconductors,” Opt. Commun. 111, 507–512 (1994).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Q. Electr.

. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Q. Electr. 37, 293-307 (2005).
[CrossRef]

Phys. Rev. E

. D. Michaelis, U. Peschel, C. Wächter, and A. Braüer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E 68, 065601(R) (2003).
[CrossRef]

. N. A. R. Bhat, and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals”. Phys. Rev. E 64, 056604 (2001).
[CrossRef]

. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002).
[CrossRef]

Other

. Slow Light: Science and Applications, J. B. Khurgin and R. S. Tucker, Eds., (CRC Press, Boca Raton, 2009).

. M. Santagiustina, “Governing the speed of light: recent advances and future perspectives of slow and fast light in microwave-photonics”, in Proc. 2009 Intern. Top. Meet. on Microwave Photonics, (Valencia, Spain, 2009) Th3.1.

. http://ab-initio.mit.edu/photons/

. R. Boyd, Nonlinear Optics Chapt. 4 (Academic Press, San Diego, 2003).

. K. Sakoda, Optical Properties of Photonic Crystals, Chapt. 2 (Springer, Berlin, 2005).

. G. P. Agrawal, Nonlinear fiber optics, Chapt. 10 (Academic Press, San Diego, 2001).

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Figures (2)

Fig. 1
Fig. 1

Comparison with model in ref. [15]. (a) Square of the group index n g 2 and SL enhancement factor S, as a function of the wavelength. In the insets the intensity of electric field |e i |2 of the Bloch mode within an elementary cell is shown at three different wavelengths. (b) Wavelength dependence of the SL scaling factor and of the effective FWM nonlinear coefficient, both normalized to their respective value at λ 11 = 1570 nm: γnorm = |γF 4(λ 1M )/γ F4(λ 11)|,Snorm =S(λ 1M )/S(λ 11).

Fig. 2
Fig. 2

(a) Comparison of the maximum achievable parametric gain coefficient, γgmP 0 (solid curves), and the actually achieved coefficient, g (dashed curves), for two different powers (P 0 = 0.9 W circles, P 0 = 1.4 W triangles) as a function of SL scaling factor S. (b) Parametric gain as a function of λ1 for a waveguide L = 1.3 mm long, for three different pump powers.

Equations (13)

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

× E ( r , ω ) = ι ω μ H ( r , ω ) , × H ( r , ω ) = ι ω ε ( r ) E ( r , ω ) ι ω P NL ( r , ω ) .
E ( r , ω ) = 1 2 i = 4 , 0 4 A i e ( r , ω i ) exp [ ι ( β i z ) ] δ i , H ( r , ω ) = 1 2 i = 4 , 0 4 A i h ( r , ω i ) exp [ ι ( β i z ) ] δ i ,
× [ e i exp ( ι β i z ) ] = ι μ ω i h i exp ( ι β i z ) , × [ h i exp ( ι β i z ) ] = ι ε ω i e i exp ( ι β i z ) .
A i z V z ^ [ e i × h i + e i × h i ] dV = j ω i V e i exp ( ι β i z ) P NL ( r , ω i ) dV
P NL ( r , ω i ) = ε 0 χ ( 3 ) ( r ; ω i ; ω i , ω i ω i ) E i E i E i + + ε 0 j = 1 , i 4 [ χ ( 3 ) ( r ; ω i ; ω j , ω j , ω i ) E j E j E i ] + ε 0 χ ( 3 ) ( r ; ω i ; ω j , ω l , ω k ) E j E l E k ,
4 a A i z = ι ω i ε 0 4 [ | A i | 2 A i V e i χ ( 3 ) e i e i e i dV + + j = 1 , i 4 | A j | 2 A i V e i χ ( 3 ) e j e j e i d V + A j A l A k exp ( ι σ i Δ β z ) V e i χ ( 3 ) e j e l e k dV ] .
v ei z ^ = 1 / 4 V ( e i × h i + e i × h i ) z ^ dV 1 / 4 V ( ε 0 ε r ( r ) | e i | 2 + μ 0 | h i | 2 ) dV = 4 a 2 V ( ε 0 ε r ( r ) | e i | 2 dV = 2 a ε 0 W i = v gi .
d A i d z = ι γ i | A i | 2 A i + 2 ι j = 1 , i 4 γ i j | A j | 2 A i + 2 ι γ Fi A l A j A k e ι σ i Δ β z , i = 1 , 2 , 3 , 4 ,
γ i = n 2 ω i a c V i ; 1 V i = n g i 2 W i 2 V ε r 3 χ xxxx ( 3 ) e i χ ( 3 ) ( r ; ω i ; ω i , ω i , ω i ) e i e i e i dV ;
γ ij = n 2 ω i a c V ij ; 1 V ij = n gi n gj W i W j V ε r 6 χ xxxx ( 3 ) e i χ ( 3 ) ( r ; ω i ; ω j , ω j , ω i ) e j e j e i dV ;
γ Fi = n 2 ω i a c V Fi ; 1 V F i = n = 1 4 ( n gn W n ) 1 / 2 V ε r 6 χ xxxx ( 3 ) e i χ ( 3 ) ( r ; ω i ; ω j , ω l , ω k ) e j e l e k dV ;
e i χ ( 3 ) ( r ) e j e l e k = m [ e im D nop χ mnop ( 3 ) ( r ; ω i ; ω j , ω l , ω k ) e jn e lo e kp ]
Δ κ = Δ β i , j = 1 2 [ γ i + 2 | i j | γ ji 2 γ 3 i 2 γ 4 i ] P i = Δ β + γ pm P 0 , g = [ γ gm 2 P 0 2 Δ κ 2 4 ] 1 / 2 ,

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