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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 11862–11868
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All-optical bistability and switching near the Dirac point of a 2-D photonic crystal.

Nadia Mattiucci, Mark J. Bloemer, and Giuseppe D’Aguanno  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11862-11868 (2013)
http://dx.doi.org/10.1364/OE.21.011862


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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

Photonic crystals (PCs) or photonic band gap structures (PBGs) [1

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

5

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. 42, 768–773 (2003).

] have opened novel venues to control light localization at the nanoscale level and to enhance light-matter interactions in an unparalleled way. Any nonlinear phenomena, such as Raman scattering, quantum-dot and quantum-well emission, nonlinear harmonic generation, all-optical switching, just to name few, can in principle benefit from the high degree of light localization achievable in these structures. Among the numerous applications of PCs, we cite photonic crystal fibers [6

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]

], photonic crystal circuits [7

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]

] and photonic crystal super-prism structures [8

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]

]. The recent discovery of graphene [9

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]

], a purely two-dimensional electronic system where the conduction band and the valence band touch each other at the Dirac point leading to remarkable electronic transport properties [10

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]

], has sparked a renewed interest in the study of Dirac points in 2-D PCs [11

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

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

]. In the case of PCs, two photonic bands touch as a pair of cones (Dirac cones) giving rise to a linear, instead of a parabolic, dispersion for photons. In a recent publication [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]

] we have studied the anomalous field localization properties in the stop band of a linear, 2-D PC made of an array of dielectric columns near its Dirac point. At normal incidence the crystal exhibits a Dirac point with 100% transmission. At angles slightly off the normal where the crystal is 100% reflective, instead of exponentially decaying fields as in a photonic stop band, the field becomes strongly localized and enhanced inside the crystal. We have explained that this anomalous localization is due to guided mode resonances (GMRs) that are the foundation of the Dirac point itself and also shape its adjacent band gap. In this work the anomalous localization properties and strong field enhancement near the Dirac point are used to boost Kerr nonlinearities and achieve switching and bistable response for low input intensities and low filling factor of the nonlinear material. In Section 2 we detail the main results of our study followed by a discussion and in Section 3 we present our conclusions.

2. Results and discussion

We start our analysis by showing in Fig. 1
Fig. 1 Cross sectional view of a 2-D PC consisting of a square array (period a) of dielectric columns with radius r and relative permittivity ε. The structure possesses a finite number N of rows along the z-direction and a total length L = Na. We suppose that the columns are made of a nonlinear Kerr-type material with a generic cubic coefficient χ(3). A plane electromagnetic wave with the electric field parallel to the axis of the columns is incident on the structure with its k-vector in the (x,z) plane (in-plane coupling) forming an angle ϑ with respect to the z-direction.
the structure under investigation.

In Fig. 2
Fig. 2 (a) Transmittance in the (ω,kx) plane. (b) Magnification of the transmittance around the Dirac point. The dashed line indicates the calculated dispersion of the Dirac-point GMR. The structure’s parameters are: N = 5, r/a = 0.2, ε = 12.5.
we show the photonic band gap structure in the (ω,kx) plane for the following parameters: 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

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]

]. The photonic band gap structure has been calculated through the transmittance, i.e. the transmitted power divided the incident power. The numerical calculation has been performed using an in-house developed code based on the Fourier-modal-method (FMM) [19

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]

].

In particular Fig. 2(a) shows the transmittance in the (ω,kx) 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 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

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 64, 16609 (2001).

]. 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]

]. The dispersion curve located in the band gap of the structure corresponds to sharp reflection resonances for close-to-normal incident conditions. As discussed at length in [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]

], the Dirac-point GMR is characterized by an exceptionally dispersive coupling strength as function of the incident angle: at close-to-normal incidence the electric field is extremely localized over the columns with localization enhancement factors exceeding 107 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
Fig. 3 Cross sectional view of the field localization in the stop-band at the Dirac-point GMR for ϑ = 0.1°. The dashed circles indicate the position of the columns.
we show, as an example, the localization of the electric field for an incident angle ϑ = 0.1°.

The field is exceptionally squeezed inside the columns with a typical, dipolar-like localization profile. Localization enhancements exceeding 104 are noted in the central row. Those features are extremely well suited to boost the nonlinear properties of the columns. In this case we suppose that the columns possess a Kerr-type cubic nonlinearity so that their local nonlinear permittivity is dependent on the local field intensity following the usual relation: εNL=ε+χ(3)|E|2. Figure 4
Fig. 4 (Left panel) Linear transmittance (red) and nonlinear transmittance (circles) with the onset of optical bistability for different incident angles along the Dirac-point GMR. The control parameter is σ = 2χ(3) Iin0c where Iin is the input intensity, ε0 the vacuum permittivity and c the speed of light in vacuo. The marks on the linear transmittance (triangle, circle, square) indicate the tuning conditions for the nonlinear calculations reported in the right panel. (Right panel) Nonlinear transmittance vs. input intensity for the incident angle reported in the left panel. Note the change in scale for the input intensity as the incident angle increases.
reports the results of the nonlinear transmittance as detailed in the figure caption. The nonlinear calculation has been performed using a mean field approach as in [21

21. P. Vicent, N. Paraire, M. Neviere, A. Koster, and R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2(7), 1106–1116 (1985). [CrossRef]

].

As expected, in agreement with the decrement of the field localization when the incident field departs from the close-to-normal condition, the threshold intensity for the onset of the optical bistability and switching increases for increasing incident angles. It is also interesting to note in the right panel how, depending on the tuning conditions on the linear resonance, the nonlinear transmittance passes from an optical limiting behavior (green curve and red curve) to a full hysteresis cycle (blue curve). To have an idea of the actual intensities needed to achieve optical bistability and switching we provide in Table 1
σChalcogenide
χ(3) = 4.4*10−20 m2/V2Silica
χ(3) = 3.4*10−22 m2/V2
10−7
3*105 W/cm2
3.9*107 W/cm2
6*10−7
1.8*106 W/cm2
2.3*108 W/cm2
2*10−56*107 W/cm27.8*109 W/cm2
a synoptic prospect which refers to the cubic nonlinearity of two widely used materials for nonlinear optical applications, namely silica and chalcogenide glasses. Silica is mainly used for temporal soliton propagation in optical fibers [22

22. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

], while chalcogenide glasses are mainly used for nanoscale photonic applications [23

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]

] due to their extremely high cubic nonlinearity and low two-photon absorption. The cubic nonlinearity of many dielectric and semiconductor materials falls in between the range set by chalcogenide glasses and silica. It is worthwhile to note that the proper material choice should be done consistently with the known limitation to all-optical switching set by material’s two-photon absorption [24

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]

].

The values of the control parameter refer to the three cases studied in the left panel of Fig. 4. For example, in the first case (σ = 10−7) the corresponding input intensity for the onset of optical bistability is ~0.3 MW/cm2 for chalcogenide glasses and ~40MW/cm2 for silica. In both cases we are speaking of extremely low intensity, well below the GW/cm2 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 w0 = λ/(πΔϑ)~200λ. This value of the beam waist is compatible, for example, with the degree of focusing necessary to obtain an intensity ~GW/cm2 from a Ti:sapphire laser at λ = 800nm.

3. Conclusions

In conclusion, we have studied all-optical switching at the Dirac-point GMR [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]

] and shown that extremely low switching intensities can be achieved even for structures in which the effective volume of the nonlinear material is remarkably small. This property may pave the way to a great variety of nonlinear devices such as: self-tunable optical sensing structures for laser eye protection, switches, optical memories, and advanced nanocircuits. In particular, the strong dependence of the Dirac-point GMR on the incident angle has applications in nonlinear angularly selective filters and collimators. Moreover, similar concepts may also be applied to harmonic generators and parametric amplifiers, which we plan to examine in future studies.

Acknowledgments

This work has been supported by DARPA SBIR project number W31P4Q-11-C-0109.

References and links

1.

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

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

3.

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

4.

J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, (Springer, 2005).

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. 42, 768–773 (2003).

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]

12.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008). [CrossRef]

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 75(6), 063813 (2007). [CrossRef]

14.

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]

15.

M. Diem, T. Koschny, and C. M. Soukoulis, “Transmission in the vicinity of the Dirac point in hexagonal photonic crystals,” Physica B 405(14), 2990–2995 (2010). [CrossRef]

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]

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]

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]

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 64, 16609 (2001).

21.

P. Vicent, N. Paraire, M. Neviere, A. Koster, and R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2(7), 1106–1116 (1985). [CrossRef]

22.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

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]

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

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987). [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals,Molding the Flow of Light. (Princeton University, 1995).
  4. J. M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals, (Springer, 2005).
  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.42, 768–773 (2003).
  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,” Nature386(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,” Nature438(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]
  12. S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A78(3), 033834 (2008). [CrossRef]
  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. A75(6), 063813 (2007). [CrossRef]
  14. 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]
  15. 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]
  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]
  17. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express20(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. B87(8), 085135 (2013). [CrossRef]
  19. 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]
  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. E64, 16609 (2001).
  21. 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]
  22. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).
  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. Express15(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]

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