OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Henry Van Driel
  • Vol. 26, Iss. 4 — Apr. 1, 2009
  • pp: 587–594

Analyzing second harmonic generation from arrays of cylinders using Dirichlet-to-Neumann maps

Lijun Yuan and Ya Yan Lu  »View Author Affiliations


JOSA B, Vol. 26, Issue 4, pp. 587-594 (2009)
http://dx.doi.org/10.1364/JOSAB.26.000587


View Full Text Article

Enhanced HTML    Acrobat PDF (302 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(190.2620) Nonlinear optics : Harmonic generation and mixing
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 30, 2008
Revised Manuscript: January 22, 2008
Manuscript Accepted: January 26, 2009
Published: March 3, 2009

Citation
Lijun Yuan and Ya Yan Lu, "Analyzing second harmonic generation from arrays of cylinders using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. B 26, 587-594 (2009)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-26-4-587


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  2. M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25-32 (1994). [CrossRef]
  3. M. Bertolotti, “Wave interations in photonic band structures: an overview,” J. Opt. A, Pure Appl. Opt. 8, S9-S32 (2006). [CrossRef]
  4. Z. Y. Ou and H. J. Kimble, “Enhanced conversion efficiency for harmonic-generation with double-resonance,” Opt. Lett. 18, 1053-1055 (1993). [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]
  9. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136-4139 (1998). [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]
  11. E. Centeno, “Second-harmonic superprism effect in photonic crystals,” Opt. Lett. 30, 1054-1056 (2005). [CrossRef] [PubMed]
  12. E. Centeno and D. Felbacq, “Second-harmonic emission in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 23, 2257-2264 (2006). [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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited