Abstract

We investigate optical parametric oscillations via four-wave mixing in a dielectric photonic crystal. Using a fully vectorial 3D time-domain approach, including both dispersion and Kerr nonlinear polarization, we analyze the response of an inverted opal. The results demonstrate the feasibility of parametric sources in isotropic media arranged in photonic band-gap geometries.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621-623 (2002).
    [CrossRef] [PubMed]
  3. A. Yariv and W. H. Louisell, �??Theory of the Optical Parametric Oscillator,�?? IEEE J. Quantum Electron. QE-2, 418-424 (1966).
    [CrossRef]
  4. B. Crosignani, P. D. Porto, and A. Yariv, �??Coupled-mode theory and slowly-varying approximation in guided-wave optics,�?? Opt. Commun. 78, 237-239 (1990).
    [CrossRef]
  5. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs N.J.,1984).
  6. E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  7. S. John, �??Strong Localization of Photons in Certain Disordered Dielectric Superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  8. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Photonic Crystals (Princeton University Press, Princeton, 1995).
  9. K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
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    [CrossRef]
  11. V. Lousse and J. P. Vigneron, �??Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,�?? Phys. Rev. E 63, 027602 (2001).
    [CrossRef]
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    [CrossRef]
  13. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, �??Optimal bistable switching in nonlinear photonic crystals,�?? Phys. Rev. E 66, 55601 (2002).
    [CrossRef]
  14. J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, �??Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,�?? IEEE J. Quantum Electron. 35, 1168-1175 (1999).
    [CrossRef]
  15. K. Srinivasan and O. Painter, �??Momentum space design of high-Q photonic crystal optical cavities,�?? Optics Express 10, 670-684 (2002). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a>
    [CrossRef] [PubMed]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego (CA), 1989).
  17. J.-P. Fève, B. Boulanger, J. Douady, �??Specific properties of cubic optical parametric interactions compared to quadratic interactions,�?? Phys. Rev. A 66, 063817 (2002).
    [CrossRef]
  18. M. Bahl, N.-C. Panoiu, and R. Osgood Jr., �??Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,�?? Phys. Rev. E 67, 56604 (2003).
    [CrossRef]
  19. Z. Y. Li and Y. Xia, �??Full vectorial model for quantum optics in three-dimensional photonic crystals,�?? Phys. Rev. A 63, 043817 (2001).
    [CrossRef]
  20. J. L. Young and R. O. Nelson, �??A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,�?? IEEE Antennas Propagat. Mag. 43, 61-77 (2001).
    [CrossRef]
  21. K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).
  22. A. Taflove and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method, 2 ed. (Artech House, London, 2000).
  23. J. Koga, �??Simulation model for the effects of nonlinear polarization on the propagation of intense pulse laser,�?? Opt. Lett. 24, 408-410 (1999).
    [CrossRef]
  24. R. W. Boyd, Nonlinear Optics, 2 ed. (Academic Press, New York, 2002).
  25. R. M. Joseph and A. Taflove, �??FDTD Maxwell�??s Equations Models for Nonlinear Electrodynamics and Optics,�?? IEEE Trans. Antennas Propagat. 45, 364-374 (1997).
    [CrossRef]
  26. S. John and K. Busch, �??Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,�?? J. Lightwave Technol. 17, 1931-1943 (1999).
    [CrossRef]
  27. A. Blanco et al., �??Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,�?? Nature 405, 437-440 (2000).
    [CrossRef] [PubMed]
  28. J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, �??The nonlinear optical properties of AlGaAs at the half band gap,�?? IEEE J. Quantum Electron. 33, 341-348 (1997).
    [CrossRef]
  29. R. Wang, X. H. Wang, B. Y. Gu, and G. Z. Yang, �??Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects,�?? Phys. Rev. B 67, 155114 (2003).
    [CrossRef]
  30. R. W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  31. M. J. A. d. Dood, A. Polmand, and J. G. Flerning, �??Modified spontaneous emission from erbium-doped photonic layer-by-layer crystals,�?? Phys. Rev. B 67, 115106 (2003).
    [CrossRef]
  32. K. Banaszek, and P. L. Knight, �??Quantum interference in three-photon down-conversion,�?? Phys. Rev. A 55, 2368
    [CrossRef]

IEEE Antennas Propagat. Mag. (1)

J. L. Young and R. O. Nelson, �??A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,�?? IEEE Antennas Propagat. Mag. 43, 61-77 (2001).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, �??The nonlinear optical properties of AlGaAs at the half band gap,�?? IEEE J. Quantum Electron. 33, 341-348 (1997).
[CrossRef]

A. Yariv and W. H. Louisell, �??Theory of the Optical Parametric Oscillator,�?? IEEE J. Quantum Electron. QE-2, 418-424 (1966).
[CrossRef]

J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, �??Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,�?? IEEE J. Quantum Electron. 35, 1168-1175 (1999).
[CrossRef]

IEEE Trans. Antennas Propagat. (2)

R. M. Joseph and A. Taflove, �??FDTD Maxwell�??s Equations Models for Nonlinear Electrodynamics and Optics,�?? IEEE Trans. Antennas Propagat. 45, 364-374 (1997).
[CrossRef]

K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).

J. Lightwave Technol. (1)

S. John and K. Busch, �??Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,�?? J. Lightwave Technol. 17, 1931-1943 (1999).
[CrossRef]

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

S. F. Mingaleev and Y. S. Kivshar, �??Nonlinear transmission and light localization in photonic-crystal waveguides,�?? J. Opt. Soc. Am. B 19, 2241-2249 (2002).
[CrossRef]

Nature (3)

S. Barland et al., �??Cavity solitons as pixels in semiconductor microcavities,�?? Nature 419, 699-702 (2002).
[CrossRef] [PubMed]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621-623 (2002).
[CrossRef] [PubMed]

A. Blanco et al., �??Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,�?? Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

B. Crosignani, P. D. Porto, and A. Yariv, �??Coupled-mode theory and slowly-varying approximation in guided-wave optics,�?? Opt. Commun. 78, 237-239 (1990).
[CrossRef]

Opt. Lett. (1)

J. Koga, �??Simulation model for the effects of nonlinear polarization on the propagation of intense pulse laser,�?? Opt. Lett. 24, 408-410 (1999).
[CrossRef]

Optics Express (1)

K. Srinivasan and O. Painter, �??Momentum space design of high-Q photonic crystal optical cavities,�?? Optics Express 10, 670-684 (2002). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a>
[CrossRef] [PubMed]

Phys. Rev. A (3)

J.-P. Fève, B. Boulanger, J. Douady, �??Specific properties of cubic optical parametric interactions compared to quadratic interactions,�?? Phys. Rev. A 66, 063817 (2002).
[CrossRef]

Z. Y. Li and Y. Xia, �??Full vectorial model for quantum optics in three-dimensional photonic crystals,�?? Phys. Rev. A 63, 043817 (2001).
[CrossRef]

K. Banaszek, and P. L. Knight, �??Quantum interference in three-photon down-conversion,�?? Phys. Rev. A 55, 2368
[CrossRef]

Phys. Rev. B (3)

M. J. A. d. Dood, A. Polmand, and J. G. Flerning, �??Modified spontaneous emission from erbium-doped photonic layer-by-layer crystals,�?? Phys. Rev. B 67, 115106 (2003).
[CrossRef]

R. Wang, X. H. Wang, B. Y. Gu, and G. Z. Yang, �??Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects,�?? Phys. Rev. B 67, 155114 (2003).
[CrossRef]

P. Tran, �??Photonic-band-structure calculation of material possessing Kerr nonlinearity,�?? Phys. Rev. B 52, 10673-10676 (1995).
[CrossRef]

Phys. Rev. E (4)

V. Lousse and J. P. Vigneron, �??Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,�?? Phys. Rev. E 63, 027602 (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, 55601 (2002).
[CrossRef]

M. Bahl, N.-C. Panoiu, and R. Osgood Jr., �??Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,�?? Phys. Rev. E 67, 56604 (2003).
[CrossRef]

R. W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, �??Strong Localization of Photons in Certain Disordered Dielectric Superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Other (6)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Photonic Crystals (Princeton University Press, Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs N.J.,1984).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego (CA), 1989).

R. W. Boyd, Nonlinear Optics, 2 ed. (Academic Press, New York, 2002).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method, 2 ed. (Artech House, London, 2000).

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Figures (3)

Fig. 1.
Fig. 1.

Spectrum of the transmitted signal for a single-cycle low-power excitation. The dotted line is the normalized input spectrum. The star identifies the wavelength used to pump the OPO. A sketch of the FCC lattice is on the upper right.

Fig. 2.
Fig. 2.

Oscillation spectrum inside the FCC for Pin =1.2MW. Excited wavelengths correspond to the peaks in Fig. 1. The star indicates the pump wavelength as launched in the PC. The insets are close-ups of the interval around 1200nm at various excitations. All graphs are normalized to the highest peak corresponding to the pump.

Fig. 3.
Fig. 3.

Oscillation spectrum at the output of the PC for Pin =1.2MW, corresponding to Fig 2: linear (blue line) and logarithmic (green line) vertical scale. The star indicates the pump wavelength.

Equations (6)

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

P th = ω 2 g Q Q + Q ,
a ± 2 = 1 4 g Q Q ± Q Δ p .
𝓔 ( ω 2 ) Δ p 2 ρ ( ω ) ( ω 2 2 Q 2 ) 2 + ( ω ω 2 ) 2 d ω ,
× E = μ 0 t H
× H = ε 0 t E + t P
t 2 P + 2 γ 0 t P + ω 0 2 f ( P ) P = ε 0 ( ε s 1 ) ω 0 2 E .

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