## All-optical bistability and switching near the Dirac point of a 2-D photonic crystal. |

Optics Express, Vol. 21, Issue 10, pp. 11862-11868 (2013)

http://dx.doi.org/10.1364/OE.21.011862

Acrobat PDF (2388 KB)

### Abstract

We investigate all-optical switching at the guided mode resonances originating near the Dirac point of a finite, 2-D photonic crystal consisting of a square lattice of dielectric columns possessing a cubic nonlinearity. The peculiar field localization properties of these Dirac-point guided mode resonances conspire to yield extremely low switching threshold at near-to-normal incidence for remarkably low filling factors of the nonlinear material.

© 2013 OSA

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**(20), 2059–2062 (1987). [CrossRef] [PubMed]

6. J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**(19), 1547–1549 (1996). [CrossRef] [PubMed]

7. J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature **386**(6621), 143–149 (1997). [CrossRef]

8. S. N. Tandon, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, and L. A. Kolodziejski, “The superprism effect using large area 2D-periodic photonic crystal slabs,” Photonics Nanostruct. Fundam. Appl. **3**(1), 10–18 (2005). [CrossRef]

9. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature **438**(7065), 197–200 (2005). [CrossRef] [PubMed]

10. A. H. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic processes of graphene,” Rev. Mod. Phys. **81**(1), 109–162 (2009). [CrossRef]

11. F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. **100**(1), 013904 (2008). [CrossRef] [PubMed]

17. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express **20**(9), 9925–9939 (2012). [CrossRef] [PubMed]

18. G. D’Aguanno, N. Mattiucci, C. Conti, and M. J. Bloemer, “Field localization and enhancement near the Dirac point of a finite defectless photonic crystal,” Phys. Rev. B **87**(8), 085135 (2013). [CrossRef]

## 2. Results and discussion

*ω*,

*k*) plane for the following parameters:

_{x}*N = 5*,

*r/*a

*= 0.2*,

*ε*= 12.5 [16

16. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. **10**(8), 582–586 (2011). [CrossRef] [PubMed]

18. G. D’Aguanno, N. Mattiucci, C. Conti, and M. J. Bloemer, “Field localization and enhancement near the Dirac point of a finite defectless photonic crystal,” Phys. Rev. B **87**(8), 085135 (2013). [CrossRef]

19. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**(5), 1024–1035 (1996). [CrossRef]

*ω*,

*k*) plane for a wide frequency range. From Fig. 2(a) it is noted the overall conical shape of both the upper and lower pass band that touch each other at the Dirac point located at ωa/2πc≅0.54. An additional lateral pass band is also present. It is important to underline that here we are dealing with a finite structure along

_{x}*z*. The pass bands are actually characterized by multiple Fabry-Perot-like transmission resonances due to the multiple interference effects of the field reflected/transmitted from each of the

*N*rows of columns the structure is made of. This is a typical characteristic of finite PCs even in lower dimensional systems such as 1-D multilayered structures [20]. Figure 2(b) shows instead a magnification of the transmittance near the Dirac point of the structure and the dispersion of the corresponding Dirac-point GMR calculated according to [18

18. G. D’Aguanno, N. Mattiucci, C. Conti, and M. J. Bloemer, “Field localization and enhancement near the Dirac point of a finite defectless photonic crystal,” Phys. Rev. B **87**(8), 085135 (2013). [CrossRef]

**87**(8), 085135 (2013). [CrossRef]

^{7}for ϑ→0°. As the incident wave departs from the close-to-normal condition the coupling strength of the GMR dramatically decreases marking the transition from localized modes to stop-band evanescent modes. In Fig. 3 we show, as an example, the localization of the electric field for an incident angle ϑ = 0.1°.

σ | Chalcogenide
χ^{(3)} = 4.4*10^{−20} m^{2}/V^{2} | Silica
χ^{(3)} = 3.4*10^{−22} m^{2}/V^{2} |
---|---|---|

10^{−7} | 3*10^{5} W/cm^{2} | 3.9*10^{7} W/cm^{2} |

6*10^{−7} | 1.8*10^{6} W/cm^{2} | 2.3*10^{8} W/cm^{2} |

2*10^{−5} | 6*10^{7} W/cm^{2} | 7.8*10^{9} W/cm^{2} |

23. V. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express **15**(15), 9205–9221 (2007). [CrossRef] [PubMed]

24. V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. **14**(20), 1140–1142 (1989). [CrossRef] [PubMed]

^{−7}) the corresponding input intensity for the onset of optical bistability is ~0.3 MW/cm

^{2}for chalcogenide glasses and ~40MW/cm

^{2}for silica. In both cases we are speaking of extremely low intensity, well below the GW/cm

^{2}range typical of nonlinear optical phenomena. These results are even more remarkable if we consider that the filling factor of the nonlinear material (ratio between the area of the column and the area of the elementary cell) is in our case only ~10%. The kind of resonances explored in this work require the use of a collimated light beam with an angular divergence Δϑ~0.1° which, roughly speaking, corresponds to a beam waist

*w*= λ/(πΔϑ)~200λ. This value of the beam waist is compatible, for example, with the degree of focusing necessary to obtain an intensity ~GW/cm

_{0}^{2}from a Ti:sapphire laser at λ = 800nm.

## 3. Conclusions

**87**(8), 085135 (2013). [CrossRef]

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, |

5. | A. Scherer, T. Yoshie, M. Loncar, J. Vuckovic, and K. Okamoto, “Photonic Crystal Nanocavities for Efficient Light Confinement and Emission,” J. Korean Phys. Soc. |

6. | J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

7. | J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature |

8. | S. N. Tandon, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, and L. A. Kolodziejski, “The superprism effect using large area 2D-periodic photonic crystal slabs,” Photonics Nanostruct. Fundam. Appl. |

9. | K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature |

10. | A. H. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic processes of graphene,” Rev. Mod. Phys. |

11. | F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. |

12. | S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A |

13. | R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A |

14. | X. Zhang, “Observing |

15. | M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B |

16. | X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. |

17. | K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express |

18. | G. D’Aguanno, N. Mattiucci, C. Conti, and M. J. Bloemer, “Field localization and enhancement near the Dirac point of a finite defectless photonic crystal,” Phys. Rev. B |

19. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

20. | G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E |

21. | P. Vicent, N. Paraire, M. Neviere, A. Koster, and R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B |

22. | Y. S. Kivshar and G. P. Agrawal, |

23. | V. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express |

24. | V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. |

**OCIS Codes**

(230.4320) Optical devices : Nonlinear optical devices

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 13, 2013

Revised Manuscript: May 2, 2013

Manuscript Accepted: May 2, 2013

Published: May 8, 2013

**Citation**

Nadia Mattiucci, Mark J. Bloemer, and Giuseppe D’Aguanno, "All-optical bistability and switching near the Dirac point of a 2-D photonic crystal.," Opt. Express **21**, 11862-11868 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11862

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### References

- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals,Molding the Flow of Light. (Princeton University, 1995).
- J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, (Springer, 2005).
- A. Scherer, T. Yoshie, M. Loncar, J. Vuckovic, and K. Okamoto, “Photonic Crystal Nanocavities for Efficient Light Confinement and Emission,” J. Korean Phys. Soc.42, 768–773 (2003).
- J. C. Knight, T. A. Birks, P. S. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett.21(19), 1547–1549 (1996). [CrossRef] [PubMed]
- J. D. Joannopoulos, P. R. Villeneuve, and S. H. Fan, “Photonic crystals: putting a new twist on light,” Nature386(6621), 143–149 (1997). [CrossRef]
- S. N. Tandon, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, and L. A. Kolodziejski, “The superprism effect using large area 2D-periodic photonic crystal slabs,” Photonics Nanostruct. Fundam. Appl.3(1), 10–18 (2005). [CrossRef]
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature438(7065), 197–200 (2005). [CrossRef] [PubMed]
- A. H. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic processes of graphene,” Rev. Mod. Phys.81(1), 109–162 (2009). [CrossRef]
- F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett.100(1), 013904 (2008). [CrossRef] [PubMed]
- S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008). [CrossRef]
- R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A75(6), 063813 (2007). [CrossRef]
- X. Zhang, “Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett.100(11), 113903 (2008). [CrossRef] [PubMed]
- M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B405(14), 2990–2995 (2010). [CrossRef]
- X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater.10(8), 582–586 (2011). [CrossRef] [PubMed]
- K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20(9), 9925–9939 (2012). [CrossRef] [PubMed]
- G. D’Aguanno, N. Mattiucci, C. Conti, and M. J. Bloemer, “Field localization and enhancement near the Dirac point of a finite defectless photonic crystal,” Phys. Rev. B87(8), 085135 (2013). [CrossRef]
- L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A13(5), 1024–1035 (1996). [CrossRef]
- G. D'Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E64, 16609 (2001).
- P. Vicent, N. Paraire, M. Neviere, A. Koster, and R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B2(7), 1106–1116 (1985). [CrossRef]
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).
- V. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express15(15), 9205–9221 (2007). [CrossRef] [PubMed]
- V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett.14(20), 1140–1142 (1989). [CrossRef] [PubMed]

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