Abstract

We develop an efficient numerical method for analyzing second harmonic generation (SHG) in two-dimensional photonic crystals composed of nonlinear circular cylinders embedded in a linear background medium. Instead of solving the governing inhomogeneous Helmholtz equation for the second harmonic wave in the entire structure directly, we define and solve a locally generated second harmonic field in each cylinder (independent of all other cylinders), then merge the field together using Dirichlet-to-Neumann (DtN) maps of the unit cells. For linear waves in a unit cell without sources, the DtN map is an operator that maps the wave field to its normal derivative on the boundary, and it can be approximated by a small matrix. A highly accurate pseudospectral method is used to solve the locally generated second harmonic wave in the cylinders. The method was applied to analyze enhanced SHG when the linear power reflectivity peaks at both the fundamental and the second harmonic frequencies.

© 2009 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
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    [CrossRef]
  3. M. Bertolotti, “Wave interations in photonic band structures: an overview,” J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
    [CrossRef]
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    [CrossRef]
  5. F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
    [CrossRef]
  6. M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
    [CrossRef]
  7. R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. 31, 3327-3329 (2006).
    [CrossRef] [PubMed]
  8. K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742-5749 (1996).
    [CrossRef]
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    [CrossRef]
  10. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387-400 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
    [CrossRef]
  14. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
    [CrossRef]
  15. J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
    [CrossRef]
  16. R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
    [CrossRef]
  17. W. Nakagawa, R. C. Tyan, and Y. Fainman, “Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation,” J. Opt. Soc. Am. A 19, 1919-1928 (2002).
    [CrossRef]
  18. A. Locatelli, D. Modotto, C. De Angelis, F. M. Pigozzo, and A. D. Capobianco, “Nonlinear bidirectional beam propagation method based on scattering operators for periodic microstructured waveguides,” J. Opt. Soc. Am. B 20, 1724-1731 (2003).
    [CrossRef]
  19. B. Maes, P. Bienstman, and R. Baets, “Modeling second-harmonic generation by use of mode expansion,” J. Opt. Soc. Am. B 22, 1378-1383 (2005).
    [CrossRef]
  20. L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for second harmonic generation in piecewise uniform waveguides,” J. Opt. Soc. Am. B 24, 2287-2293 (2007).
    [CrossRef]
  21. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann Maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
    [CrossRef]
  22. L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
    [CrossRef]
  23. G. Bao and Y. Chen, “A nonlinear grating problem in diffractive optics,” SIAM J. Math. Anal. 28, 322-337 (1997).
    [CrossRef]
  24. G. BaoZ. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]
  25. Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).
  26. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
    [CrossRef]
  27. L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
    [CrossRef]
  28. Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
    [CrossRef]
  29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

2008

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
[CrossRef]

2007

L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for second harmonic generation in piecewise uniform waveguides,” J. Opt. Soc. Am. B 24, 2287-2293 (2007).
[CrossRef]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

2006

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
[CrossRef]

M. Bertolotti, “Wave interations in photonic band structures: an overview,” J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
[CrossRef]

E. Centeno and D. Felbacq, “Second-harmonic emission in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 23, 2257-2264 (2006).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann Maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. 31, 3327-3329 (2006).
[CrossRef] [PubMed]

2005

2004

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

2003

2002

2000

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387-400 (2000).
[CrossRef]

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
[CrossRef]

1998

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

1997

G. Bao and Y. Chen, “A nonlinear grating problem in diffractive optics,” SIAM J. Math. Anal. 28, 322-337 (1997).
[CrossRef]

1996

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

1995

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

1994

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25-32 (1994).
[CrossRef]

1993

Andreani, L. C.

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

Arie, A.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Baets, R.

Bahabad, A.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Bao, G.

G. BaoZ. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
[CrossRef]

G. Bao and Y. Chen, “A nonlinear grating problem in diffractive optics,” SIAM J. Math. Anal. 28, 322-337 (1997).
[CrossRef]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

Bertolotti, M.

M. Bertolotti, “Wave interations in photonic band structures: an overview,” J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
[CrossRef]

Bienstman, P.

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Capobianco, A. D.

Centeno, E.

Chen, J.

Chen, Y.

G. Bao and Y. Chen, “A nonlinear grating problem in diffractive optics,” SIAM J. Math. Anal. 28, 322-337 (1997).
[CrossRef]

Chen, Z. M.

Cheng, C.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

De Angelis, C.

Ding, J.

Dumeige, Y.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Fainman, Y.

Fan, Y. X.

Fejer, M. M.

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25-32 (1994).
[CrossRef]

Felbacq, D.

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Hirao, K.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Huang, Y.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann Maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kimble, H. J.

Lederer, F.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Lee, R. K.

Letartre, X.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Levenson, A.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Li, J. J.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

Li, R.

R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. 31, 3327-3329 (2006).
[CrossRef] [PubMed]

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Li, Z. Y.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

Liscidini, M.

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

Locatelli, A.

Lu, Y. Y.

Maes, B.

Marinica, D. C.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Modotto, D.

Nakagawa, W.

Ohtaka, K.

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

Ou, Z. Y.

Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Pigozzo, F. M.

Qiu, J. R.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Raineri, F.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Ren, F. F.

R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. 31, 3327-3329 (2006).
[CrossRef] [PubMed]

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Sakoda, K.

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

Shabanov, S. V.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Si, J. H.

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
[CrossRef]

Tyan, R. C.

Wang, H. T.

R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. 31, 3327-3329 (2006).
[CrossRef] [PubMed]

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, H. J.

Wu, Y.

Xu, Q.

Xu, Y.

Yariv, A.

Yuan, L.

L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for second harmonic generation in piecewise uniform waveguides,” J. Opt. Soc. Am. B 24, 2287-2293 (2007).
[CrossRef]

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
[CrossRef]

Zhang, D. Z.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

IEEE Photon. Technol. Lett.

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
[CrossRef]

J. Comput. Math.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

J. Lightwave Technol.

J. Opt. A, Pure Appl. Opt.

M. Bertolotti, “Wave interations in photonic band structures: an overview,” J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi-phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Phys. Rev. B

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77, 195127 (2008).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B 77, 115124 (2008).
[CrossRef]

F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. R. Qiu, J. H. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B 70, 245109 (2004).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742-5749 (1996).
[CrossRef]

Phys. Rev. E

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. E 68, 066617 (2003).
[CrossRef]

Phys. Rev. Lett.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

Phys. Today

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25-32 (1994).
[CrossRef]

SIAM J. Math. Anal.

G. Bao and Y. Chen, “A nonlinear grating problem in diffractive optics,” SIAM J. Math. Anal. 28, 322-337 (1997).
[CrossRef]

Other

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

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Figures (6)

Fig. 1
Fig. 1

Linear reflection spectrum for one array of cylinders, where the incident wave has a fixed angle of incidence θ 0 = 8.25 ° . The vertical axis is the power reflectivity defined in Eq. (43).

Fig. 2
Fig. 2

Frequency dependence of the normalized reflected power of the second harmonic wave, defined in Eq. (44), for one array of cylinders. The angle of incidence of the pump wave is fixed at θ 0 = 8.25 ° .

Fig. 3
Fig. 3

Magnitudes of the fundamental frequency wave (left) and the second harmonic wave (right) for one array of cylinders at γ = 0.8714 . The incident wave has a magnitude of 10 6 V m and an angle of incidence θ 0 = 8.25 ° .

Fig. 4
Fig. 4

Linear reflection spectrum for five arrays of cylinders, where the incident wave has a fixed angle of incidence θ 0 = 7.6 ° . The vertical axis is the power reflectivity defined in Eq. (43).

Fig. 5
Fig. 5

Frequency dependence of the normalized reflected power of the second harmonic wave, defined in Eq. (44), for five arrays of cylinders. The angle of incidence of the pump wave is fixed at θ 0 = 7.6 ° .

Fig. 6
Fig. 6

Magnitudes of the fundamental frequency wave (left) and the second harmonic wave (right) for five arrays of cylinders at γ = 0.86943 . The incident wave has a magnitude of 10 6 V m and an angle of incidence θ 0 = 7.6 ° .

Equations (58)

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

x 2 U + y 2 U + k 0 2 n 2 ( x ; ω ) U = k 0 2 χ ( 2 ) ( x ; ω ) U ¯ V ,
x 2 V + y 2 V + 4 k 0 2 n 2 ( x ; 2 ω ) V = 2 k 0 2 χ ( 2 ) ( x ; 2 ω ) U 2 ,
U ( i ) ( x ) = E 0 exp { i [ α 0 ( 1 ) x β 0 ( 1 ) ( y d ) ] } ,
α 0 ( 1 ) = k 0 n 0 sin ( θ 0 ) , β 0 ( 1 ) = k 0 n 0 cos ( θ 0 ) .
U ( t ) ( x ) = E 0 j = + T j ( 1 ) exp { i [ α j ( 1 ) x β j ( 1 ) y ] } , y < 0 ,
U ( r ) ( x ) = E 0 j = + R j ( 1 ) exp { i [ α j ( 1 ) x + β j ( 1 ) y ] } , y > d ,
V ( t ) ( x ) = E 0 j = + T j ( 2 ) exp { i [ α j ( 2 ) x β j ( 2 ) y ] } , y < 0 ,
V ( r ) ( x ) = E 0 j = + R j ( 2 ) exp { i [ α j ( 2 ) x + β j ( 2 ) y ] } , y > d ,
α j ( 1 ) = α 0 ( 1 ) + 2 j π a , β j ( 1 ) = k 0 2 n 0 2 [ α j ( 1 ) ] 2 ,
α j ( 2 ) = 2 α 0 ( 1 ) + 2 j π a , β j ( 2 ) = 4 k 0 2 n 0 2 [ α j ( 2 ) ] 2 .
y U i S 1 U = 2 i β 0 ( 1 ) E 0 exp { i α 0 ( 1 ) x } , y = d ,
y U + i S 1 U = 0 , y = 0 ,
y V i S 2 V = 0 , y = d ,
y V + i S 2 V = 0 , y = 0 .
U ( x + a , y ) = μ U ( x ) , x U ( x + a , y ) = μ x U ( x ) ,
V ( x + a , y ) = μ 2 V ( x ) , x V ( x + a , y ) = μ 2 x V ( x ) ,
x 2 U + y 2 U + k 0 2 n 2 ( x ; ω ) U = 0 ,
Q j V j = y V j φ j , Y j V j = V 0 + ϕ j ,
Q 0 = i S 2 , Y 0 = I , φ 0 = 0 , ϕ 0 = 0 ,
( Q m i S 2 ) V m = φ m , V 0 = Y m V m ϕ m ,
x 2 W + y 2 W + 4 k 0 2 n 2 ( x ; 2 ω ) W = 2 k 0 2 χ ( 2 ) ( x ; 2 ω ) U 2 in D
V v = Λ V + f on Ω j
[ x V 0 j y V j + 1 x V 1 j y V j ] = [ Λ 11 Λ 12 Λ 13 Λ 14 Λ 21 Λ 22 Λ 23 Λ 24 Λ 31 Λ 32 Λ 33 Λ 34 Λ 41 Λ 42 Λ 43 Λ 44 ] [ V 0 j V j + 1 V 1 j V j ] + [ f 1 f 2 f 3 f 4 ] ,
y [ V j + 1 V j ] = M [ V j + 1 V j ] + [ g 1 g 2 ] ,
C 1 = Λ 21 + μ 2 Λ 23 , C 2 = Λ 41 + μ 2 Λ 43 ,
D 0 = Λ 31 + μ 2 Λ 33 μ 2 Λ 11 μ 4 Λ 13 ,
D 1 = D 0 1 ( μ 2 Λ 12 Λ 32 ) , D 2 = D 0 1 ( μ 2 Λ 14 Λ 34 ) ,
M = [ M 11 M 12 M 21 M 22 ] = [ Λ 22 + C 1 D 1 Λ 24 + C 1 D 2 Λ 42 + C 2 D 1 Λ 44 + C 2 D 2 ] ,
g 1 = C 1 D 0 1 ( μ 2 f 1 f 3 ) + f 2 ,
g 2 = C 2 D 0 1 ( μ 2 f 1 f 3 ) + f 4 .
Z = ( Q j M 22 ) 1 M 21 ,
h = ( Q j M 22 ) 1 ( g 2 φ j ) ,
Q j + 1 = M 11 + M 12 Z ,
Y j + 1 = Y j Z ,
φ j + 1 = M 12 h + g 1 ,
ϕ j + 1 = ϕ j Y j h .
x 2 V + y 2 V + 4 k 0 2 n 1 2 V = 2 k 0 2 χ ( 2 ) U 2 , x D ,
x 2 V + y 2 V + 4 k 0 2 n 2 2 V = 0 , x D .
[ v V D v V Ω ] = A [ V D V Ω ] = [ A 11 A 12 A 21 A 22 ] [ V D V Ω ] ,
x 2 S + y 2 S + 4 k 0 2 n 1 2 S = 0 in D .
x 2 W + y 2 W + 4 k 0 2 n 1 2 W = 2 k 0 2 χ ( 2 ) U 2 in D
v S D = B S D
v V D = v S D + v W D , S D = V D .
Λ = A 21 ( B A 11 ) 1 A 12 + A 22 ,
f = A 21 ( B A 11 ) 1 v W D .
r 2 2 W r 2 + r W r + 2 W θ 2 + 4 r 2 k 0 2 n 1 2 W = 2 r 2 k 0 2 χ ( 2 ) U 2
W ( r , θ ) = W ( r , θ ̃ ) , for δ < r < 0 and θ ̃ = ( θ + π ) mod ( 2 π ) .
C = [ c 00 c ̃ 0 c 0 q C ̂ c q 0 c ̃ q c q q ] ,
( R 2 D ̂ + R C ̂ + 4 k 0 2 n 1 2 R 2 ) W + d 2 W d θ 2 = 2 k 0 2 χ ( 2 ) R 2 F ,
W = [ W ( r 1 , θ ) W ( r 2 , θ ) W ( r q 1 , θ ) ] , F = [ U 2 ( r 1 , θ ) U 2 ( r 2 , θ ) U 2 ( r q 1 , θ ) ] ,
R = [ r 1 r 2 r q 1 ] .
R 2 D ̂ + R C ̂ + 4 k 0 2 n 1 2 R 2 = P [ λ 1 λ 2 λ q 1 ] P 1 ,
d 2 ψ k d θ 2 + λ k ψ k = 2 k 0 2 χ ( 2 ) τ k ( θ ) , 0 θ < 2 π
P ψ = W , P τ = F .
r W r = δ = c ̃ q W = b ψ ,
r W ( δ , θ ) = k = 1 ( q 1 ) 2 b k ψ q k ( θ ̃ ) + k = ( q 1 ) 2 + 1 q 1 b k ψ k ( θ ) .
P.R. = Re [ j β j ( 1 ) β 0 ( 1 ) R j ( 1 ) 2 ] ,
P.R. ( 2 ) = Re [ j β j ( 2 ) β 0 ( 1 ) R j ( 2 ) 2 ] .

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