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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11952–11964

Efficient numerical method for analyzing optical bistability in photonic crystal microcavities

Lijun Yuan and Ya Yan Lu  »View Author Affiliations

Optics Express, Vol. 21, Issue 10, pp. 11952-11964 (2013)

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Nonlinear optical effects can be enhanced by photonic crystal microcavities and be used to develop practical ultra-compact optical devices with low power requirements. The finite-difference time-domain method is the standard numerical method for simulating nonlinear optical devices, but it has limitations in terms of accuracy and efficiency. In this paper, a rigorous and efficient frequency-domain numerical method is developed for analyzing nonlinear optical devices where the nonlinear effect is concentrated in the microcavities. The method replaces the linear problem outside the microcavities by a rigorous and numerically computed boundary condition, then solves the nonlinear problem iteratively in a small region around the microcavities. Convergence of the iterative method is much easier to achieve since the size of the problem is significantly reduced. The method is presented for a specific two-dimensional photonic crystal waveguide-cavity system with a Kerr nonlinearity, using numerical methods that can take advantage of the geometric features of the structure. The method is able to calculate multiple solutions exhibiting the optical bistability phenomenon in the strongly nonlinear regime.

© 2013 OSA

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(190.1450) Nonlinear optics : Bistability
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

Original Manuscript: February 15, 2013
Revised Manuscript: April 12, 2013
Manuscript Accepted: April 24, 2013
Published: May 8, 2013

Lijun Yuan and Ya Yan Lu, "Efficient numerical method for analyzing optical bistability in photonic crystal microcavities," Opt. Express 21, 11952-11964 (2013)

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  1. H. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).
  2. C. M. Bowden and A. M. Zheltikov, “Nonlinear optics of photonic crystals,” J. Opt. Soc. Am. B19,2046–2048 (2002). [CrossRef]
  3. R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).
  4. K. J. Vahala, “Optical microcavities,” Nature424,839–846 (2003). [CrossRef] [PubMed]
  5. M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3,211–219 (2004). [CrossRef] [PubMed]
  6. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhance nonlinear optics in photonic-crystal microcavities,” Opt. Express15,16161–16176 (2007). [CrossRef] [PubMed]
  7. G. S. Agarwal and S. D. Gupta, “Effect of nonlinear boundary conditions on nonlinear phenomena in optical resonators,” Opt. Lett.12,829–831 (1987). [CrossRef] [PubMed]
  8. J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B44,8214–8225 (1991). [CrossRef]
  9. M. Midrio, “Shooting technique for the computation of the plane-wave reflection and transmission through one-dimensional nonlinear inhomogeneous dielectric structres,” J. Opt. Soc. Am. B18,1866–1871 (2001). [CrossRef]
  10. A. Suryanto, E. van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron.35,313–332 (2003). [CrossRef]
  11. A. Suryanto, E. van Groesen, and M. Hammer, “Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with defect,” J. Nonlinear Opt. Phy. Mater.12,187–204 (2003). [CrossRef]
  12. P. K. Kwan and Y. Y. Lu, “Computing optical bistability in one-dimensional nonlinear structures,” Opt. Commun.238,169–175 (2004). [CrossRef]
  13. A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
  14. G. Baruch, G. Fibich, and S. Tsynkov, “A high-order numerical method for the nonlinenar Helmholtz equation in multi-dimensional layered media,” J. Comput. Phys.228,3789–3815 (2009). [CrossRef]
  15. Z. Xu and G. Bao, “A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects,” J. Opt. Soc. Am. A27,2347–2353 (2010). [CrossRef]
  16. E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B62,R7683–R7686 (2000). [CrossRef]
  17. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B19,2241–2249 (2002). [CrossRef]
  18. J. Bravo-Abad, S. Fan, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightw. Technol.25,2539–2546 (2007). [CrossRef]
  19. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express16,17383–17399 (2008). [CrossRef] [PubMed]
  20. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  21. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B24,2860–2867 (2007). [CrossRef]
  22. S. Li and Y. Y. Lu, “Efficient method for analyzing leaky cavities in two-dimensional photonic crystals,” J. Opt. Soc. Am. B26,2427–2433 (2009). [CrossRef]
  23. Z. Hu and Y. Y. Lu, “A simple boundary condition for terminating photonic crystal waveguides,” J. Opt. Soc. Am. B29,1356–1360 (2012). [CrossRef]
  24. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol.24,3448–3453 (2006). [CrossRef]
  25. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A23,3217–3222 (2006). [CrossRef]
  26. L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000). [CrossRef]
  27. L. Yuan and Y. Y. Lu, “Analyzing second harmonic generation from arrays of cylinders using the Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B26,587–594 (2009). [CrossRef]

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