## Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs

Optics Express, Vol. 18, Issue 13, pp. 14301-14310 (2010)

http://dx.doi.org/10.1364/OE.18.014301

Acrobat PDF (3053 KB)

### Abstract

We discuss the simultaneous existence of phononic and photonic band gaps in a periodic array of holes drilled in a Si membrane. We investigate in detail both the centered square lattice and the boron nitride (BN) lattice with two atoms per unit cell which include the simple square, triangular and honeycomb lattices as particular cases. We show that complete phononic and photonic band gaps can be obtained from the honeycomb lattice as well as BN lattices close to honeycomb. Otherwise, all investigated structures present the possibility of a complete phononic gap together with a photonic band gap of a given symmetry, odd or even, depending on the geometrical parameters.

© 2010 OSA

## 1. Introduction

15. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B **10**(2), 283 (1993). [CrossRef]

16. S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express **3**(1), 4–11 (1998). [CrossRef] [PubMed]

17. S. Shi, C. Chen, and D. W. Prather, “Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers,” J. Opt. Soc. Am. A **21**(9), 1769 (2004). [CrossRef]

19. T.-I. Weng and G. Y. Guo, “Band structure of honeycomb photonic crystal slabs,” J. Appl. Phys. **99**(9), 093102 (2006). [CrossRef]

20. M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. **89**(22), 227402 (2002). [CrossRef] [PubMed]

21. P. Lacharmoise, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Optical cavity enhancement of light–sound interaction in acoustic phonon cavities,” Appl. Phys. Lett. **84**(17), 3274 (2004). [CrossRef]

22. M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. **88**(25), 251907 (2006). [CrossRef]

25. S. Mohammadi, A. A. Eftekhar, and A. Adibi, “Large Simultaneous Band Gaps for Photonic and Phononic Crystal Slabs,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, p. CFY1 (Optical Society of America, 2008). http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2008-CFY1.

22. M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. **88**(25), 251907 (2006). [CrossRef]

24. S. Sadat-Saleh, S. Benchabane, F. I. Baida, M. P. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. **106**(7), 074912 (2009). [CrossRef]

22. M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. **88**(25), 251907 (2006). [CrossRef]

23. M. Maldovan and E. L. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B **83**(4), 595–600 (2006). [CrossRef]

24. S. Sadat-Saleh, S. Benchabane, F. I. Baida, M. P. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. **106**(7), 074912 (2009). [CrossRef]

26. A. V. Akimov, Y. Tanaka, A. B. Pevtsov, S. F. Kaplan, V. G. Golubev, S. Tamura, D. R. Yakovlev, and M. Bayer, “Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials,” Phys. Rev. Lett. **101**(3), 033902 (2008). [CrossRef] [PubMed]

8. J. C. Hsu and T. T. Wu, “Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates,” Phys. Rev. B **74**(14), 144303 (2006). [CrossRef]

14. T. T. Wu, Z. G. Huang, T.-C. Tsai, and T. C. Wu, “Evidence of complete band gap and resonances in a plate with periodic stubbed surface,” Appl. Phys. Lett. **93**(11), 111902 (2008). [CrossRef]

## 2. Geometry and method of calculation

_{Si}of the Si slab, the filling fraction f and the ratio α = r

_{1}/r

_{2}of the radii of the two types of holes in the unit cell. The filling fraction of the air holes in the membrane is given by:for the square arrangement and by:for the BN structure. The more common single square lattice is obtained in the structure of Fig. 1(b) for α = 0. The boron nitride lattice, depicted in Fig. 1(c), includes the triangular (α = 0) and the honeycomb (α = 1) arrays as particular cases.

12. J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A. C. Hladky-Hennion, “Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates,” Phys. Rev. B **77**(8), 085415 (2008). [CrossRef]

_{Si}/a from 0.4 to 0.7, f from 0.3 to 0.7 and α from 0 to 1). Then, we search for the photonic band gaps (either complete or for one type of symmetry) in the same ranges of parameters. In general, the complete photonic band gaps occur only in a few cases which are difficult to obtain for all symmetries. Therefore, the full acoustic and optical band gap can be obtained in many situations with a complete phononic gap together with a photonic gap of a given (odd or even) symmetry. It is worth mentioning that in a real photonic crystal device the excitation of even or odd modes separately can be easily achieved by a proper selection of the polarization of the injected light. This means that, in principle, a photonic band gap occurring for a unique symmetry (odd or even) should be enough for most functionalities such as cavities, waveguides or splitters. Let us also mention that in the slab geometry, the photonic gaps have to be searched only below the light cone in vacuum. However, these gaps should preferably occur at dimensionless frequencies Ω below 0.5, otherwise they will be restricted only to a very small area of the Brillouin zone close to the light cone and are therefore not very interesting.

_{air}) can be reduced to a small slab since elastic waves cannot obviously propagate in vacuum. The air is modeled as a low impedance medium with very low density and very high velocity of sound. The convergence of the calculation is quite fast [12

12. J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A. C. Hladky-Hennion, “Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates,” Phys. Rev. B **77**(8), 085415 (2008). [CrossRef]

_{Si}/a = 0.6 and r/a = 0.43.

_{air}/a>3.4 while the branches over the light cone still continue to move to lower frequencies. One mention also that the number of plane waves have been adapted to each thickness of the unit cell to achieve a good convergence (2673 for h

_{air}/a equal to 1.4, then 3645 and 7047 for h

_{air}/a equal to 3.4 and 7.4 respectively). In the rest of the paper, the air thickness has been chosen equal to h

_{air}/a = 7.4 to insure the stability of the whole branches under the light cone and the calculations have been performed with a number of plane waves equal to 7047. In Fig. 3 , the previous PWE result [Fig. 3(b)] is compared to the dispersion curves calculated on the same photonic structure using another numerical method, i.e. the layered multiple scattering (LMS) method [27

27. G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particules,” Phys. Rev. B **73**(3), 035115 (2006). [CrossRef]

28. N. Stefanou, V. Yannopapas, and A. Modinos, “A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. **132**(1-2), 189–196 (2000). [CrossRef]

## 3. Square lattice

_{1}= 0, r

_{2}= r, α = 0). Figure 4 reports the evolution of both phononic and photonic gaps for each symmetry, even (red) and odd (blue), as a function of the filling factor f and for a set of silicon plate thicknesses h

_{Si}/a in the range [0.4, 0.7].

_{Si}/a = 0.4 and for a high value of the filling factor f = 0.7 (black vertical arrow). Unfortunately, from the photonic side, this gap appears in a very restricted region of the Brillouin zone (Ω = [0.553, 0.658]), near the M point, just below the light cone. It means that this solution is not really interesting. To cover the full directions of the Brillouin zone, the reduced frequency value should be lower than 0.5. With this condition, there is no overlap between the photonic gaps of both symmetries.

_{Si}/a = [0.5, 0.6] and for filling factors f≥0.6. In this thickness range, there are photonic gaps (either even or odd) at frequencies below 0.5.

_{Si}/a = 0.6, f = 0.65, r/a = 0.455) (see the black vertical dotted line in Fig. 4). In view of telecom applications, the photonic band gap wavelength has to be chosen close to 1550nm. Then, the actual geometrical parameters become a = 701nm, h

_{Si}= 421nm and r = 315nm for the odd gap and a = 590nm, h

_{Si}= 350nm and r = 265nm for the even gap. For these structures, the mid-gap acoustic frequency falls at 4.2GHz and 5.0GHz respectively. With these parameters, the separation between neighboring holes becomes respectively 70nm and 60nm and makes this periodic crystal geometry technologically realizable.

_{2}/r

_{1}, for a thickness h

_{Si}/a = 0.6 and a filling factor f = 0.65. It is worth noticing that no more favorable situations can be found when the square lattice contains two different cylinders in the unit cell.

## 4. Honeycomb lattice

_{1}= r

_{2}= r, α = 1), the evolution of the phononic and photonic band gaps for each symmetry as a function of the filling factor f and for different values of the thickness of the silicon plate h

_{Si}/a.

_{Si}/a from 0.4 to 0.7. The odd gaps are in general included in the even gaps except at low filling factor. The limitation comes this time from the photonic side. For the latter, the odd gap exists in the full range of the filling factor and for all the studied values of h

_{Si}/a whereas the even gap is present for h

_{Si}/a≤0.5 and progressively closes when the filling factor increases. Nevertheless, a complete phononic and photonic band gap, represented with a grey area in Fig. 6, occurs provided the thickness of the slab is in the range h

_{Si}/a = [0.4-0.5]. Assuming that the dimensionless photonic frequency gap should be lower than 0.5 to cover all directions of the Brillouin zone, one can define as an example a set of parameters (h

_{Si}/a = 0.5, f = 0.45, r/a = 0.249) (black vertical dashed line) which leads to a complete phononic and photonic band gap. Then, by assuming that the photonic midgap occurs at the telecommunication wavelength of 1550nm, we find the following geometrical parameters: a = 687nm, h

_{Si}= 330nm and the hole radius r = 171nm. The separation between neighboring holes is then 55nm, which is quite acceptable for the technological fabrication of the sample. With this lattice parameter, the phononic mid-gap frequency occurs at 4.9GHz.

_{Si}/a≤0.5 and can be chosen for several filling factors. As an example, the reduced parameters (h

_{Si}/a = 0.4, f = 0.45, r/a = 0.249) lead to a band gap of even photonic symmetry (red vertical dashed line). One can also design structures with an odd photonic gap provided the thickness of the slab h

_{Si}/a≥0.5 in order to insure the dimensionless frequency be lower than 0.5. Many filling factors, higher than 0.35, are suitable and one example is given by the following parameters (h

_{Si}/a = 0.7, f = 0.45, r/a = 0.249) (blue vertical dashed line).

## 5. Boron nitride lattice

_{Si}/a = 0.5 and h

_{Si}/a = 0.6 and for a filling factor f = 0.45.

_{Si}/a≥0.6, an odd gap can appear for all BN lattices (from triangular to honeycomb) whereas the even gaps remain open only towards the honeycomb lattice. In the photonic side, an odd gap exists for all BN lattices. The largest gaps of even symmetry occur towards the triangular lattice at frequencies around or below Ω = 0.4. Nevertheless, the even modes can also display a narrow gap towards the honeycomb lattice (when α>0.8) provided the thickness of the slab is relatively small (h

_{Si}/a≤0.5). In the latter case, this gap is included inside the odd one and gives rise to a complete phononic and photonic band gap, as already discussed in the case of the honeycomb lattice. No more complete photonic band gaps are found from the BN structures.

_{Si}/a = 0.5 in order to keep the odd gap in the frequency range below Ω = 0.5 (otherwise the gap occurs only in a very restricted range of the Brillouin zone close to the light cone). Finally, a sufficiently wide gap of even symmetry requires α≤0.8.

_{Si}/a = 0.6, f = 0.45, α = 0.6) (see the black vertical dotted line in Fig. 7). In telecom applications, the corresponding structure can be defined with the geometrical parameters a = 637nm, h

_{Si}= 382nm, r

_{1}= 115nm and r

_{2}= 192nm for the odd photonic modes and a = 491nm, h

_{Si}= 295nm, r

_{1}= 89nm and r

_{2}= 148nm for the even one. These lattice parameters lead to phononic mid-gap frequencies of 5.15GHz and 6.68GHz respectively.

## 6. Conclusion

**88**(25), 251907 (2006). [CrossRef]

24. S. Sadat-Saleh, S. Benchabane, F. I. Baida, M. P. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. **106**(7), 074912 (2009). [CrossRef]

## Acknowledgement

## References and links

1. | M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. |

2. | M. M. Sigalas and E. N. Economou, “Band structure of elastic waves in two dimensional systems,” Solid State Commun. |

3. | For a comprehensive list of references on phononic crystals, see the phononic database at http://www.univ-lehavre.fr/recherche/lomc/phonon/PhononicDatabase1.html. |

4. | Y. Pennec, B. Djafari-Rouhani, J. O. Vasseur, H. Larabi, A. Khelif, A. Choujaa, S. Benchabane, and V. Laude, “Acoustic channel drop tunneling in a phononic crystal,” Appl. Phys. Lett. |

5. | Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally resonant sonic materials,” Science |

6. | J. Bucay, E. Roussel, J. O. Vasseur, P. A. Deymier, A.-C. Hladky-Hennion, Y. Pennec, K. Muralidharan, B. Djafari-Rouhani, and B. Dubus, “Positive, negative, zero refraction, and beam splitting in a solid/air phononic crystal: Theoretical and experimental study,” Phys. Rev. B |

7. | L. Fok, M. Ambati, and Z. Xiang, “Acoustic Metamaterials,” MRS Bull. |

8. | J. C. Hsu and T. T. Wu, “Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates,” Phys. Rev. B |

9. | A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

10. | J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, and Y. Pennec, “Absolute band gaps in two-dimensional phononic crystal plates,” in Proceeding of IMECE 2006, ASME International Mechanical Engineering Congress and Exhibition, Chicago, Illinois, (5–10 Nov. 2006), pp13353. |

11. | C. Charles, B. Bonello, and F. Ganot, “Propagation of guided elastic waves in 2D phononic crystals,” Ultrasonics |

12. | J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A. C. Hladky-Hennion, “Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates,” Phys. Rev. B |

13. | Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur, and A. C. Hladky-Hennion, “Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate,” Phys. Rev. B |

14. | T. T. Wu, Z. G. Huang, T.-C. Tsai, and T. C. Wu, “Evidence of complete band gap and resonances in a plate with periodic stubbed surface,” Appl. Phys. Lett. |

15. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

16. | S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express |

17. | S. Shi, C. Chen, and D. W. Prather, “Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers,” J. Opt. Soc. Am. A |

18. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

19. | T.-I. Weng and G. Y. Guo, “Band structure of honeycomb photonic crystal slabs,” J. Appl. Phys. |

20. | M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. |

21. | P. Lacharmoise, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Optical cavity enhancement of light–sound interaction in acoustic phonon cavities,” Appl. Phys. Lett. |

22. | M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. |

23. | M. Maldovan and E. L. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B |

24. | S. Sadat-Saleh, S. Benchabane, F. I. Baida, M. P. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. |

25. | S. Mohammadi, A. A. Eftekhar, and A. Adibi, “Large Simultaneous Band Gaps for Photonic and Phononic Crystal Slabs,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, p. CFY1 (Optical Society of America, 2008). http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2008-CFY1. |

26. | A. V. Akimov, Y. Tanaka, A. B. Pevtsov, S. F. Kaplan, V. G. Golubev, S. Tamura, D. R. Yakovlev, and M. Bayer, “Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials,” Phys. Rev. Lett. |

27. | G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particules,” Phys. Rev. B |

28. | N. Stefanou, V. Yannopapas, and A. Modinos, “A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(130.3120) Integrated optics : Integrated optics devices

(230.0230) Optical devices : Optical devices

(230.1040) Optical devices : Acousto-optical devices

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 1, 2010

Revised Manuscript: May 13, 2010

Manuscript Accepted: May 15, 2010

Published: June 18, 2010

**Citation**

Y. Pennec, B. Djafari Rouhani, E. H. El Boudouti, C. Li, Y. El Hassouani, J. O. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, and A. Martinez, "Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs," Opt. Express **18**, 14301-14310 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14301

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### References

- M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. 71(13), 2022–2025 (1993). [CrossRef] [PubMed]
- M. M. Sigalas and E. N. Economou, “Band structure of elastic waves in two dimensional systems,” Solid State Commun. 86(3), 141–143 (1993). [CrossRef]
- For a comprehensive list of references on phononic crystals, see the phononic database at http://www.univ-lehavre.fr/recherche/lomc/phonon/PhononicDatabase1.html .
- Y. Pennec, B. Djafari-Rouhani, J. O. Vasseur, H. Larabi, A. Khelif, A. Choujaa, S. Benchabane, and V. Laude, “Acoustic channel drop tunneling in a phononic crystal,” Appl. Phys. Lett. 87(26), 261912 (2005). [CrossRef]
- Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally resonant sonic materials,” Science 289(5485), 1734–1736 (2000). [CrossRef] [PubMed]
- J. Bucay, E. Roussel, J. O. Vasseur, P. A. Deymier, A.-C. Hladky-Hennion, Y. Pennec, K. Muralidharan, B. Djafari-Rouhani, and B. Dubus, “Positive, negative, zero refraction, and beam splitting in a solid/air phononic crystal: Theoretical and experimental study,” Phys. Rev. B 79(21), 214305 (2009). [CrossRef]
- L. Fok, M. Ambati, and Z. Xiang, “Acoustic Metamaterials,” MRS Bull. 33, 931 (2008). [CrossRef]
- J. C. Hsu and T. T. Wu, “Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates,” Phys. Rev. B 74(14), 144303 (2006). [CrossRef]
- A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, and V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046610 (2006). [CrossRef] [PubMed]
- J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, and Y. Pennec, “Absolute band gaps in two-dimensional phononic crystal plates,” in Proceeding of IMECE 2006, ASME International Mechanical Engineering Congress and Exhibition, Chicago, Illinois, (5–10 Nov. 2006), pp13353.
- C. Charles, B. Bonello, and F. Ganot, “Propagation of guided elastic waves in 2D phononic crystals,” Ultrasonics 44, e1209 (2006). [CrossRef] [PubMed]
- J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A. C. Hladky-Hennion, “Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates,” Phys. Rev. B 77(8), 085415 (2008). [CrossRef]
- Y. Pennec, B. Djafari-Rouhani, H. Larabi, J. O. Vasseur, and A. C. Hladky-Hennion, “Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate,” Phys. Rev. B 78(10), 104105 (2008). [CrossRef]
- T. T. Wu, Z. G. Huang, T.-C. Tsai, and T. C. Wu, “Evidence of complete band gap and resonances in a plate with periodic stubbed surface,” Appl. Phys. Lett. 93(11), 111902 (2008). [CrossRef]
- E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10(2), 283 (1993). [CrossRef]
- S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3(1), 4–11 (1998). [CrossRef] [PubMed]
- S. Shi, C. Chen, and D. W. Prather, “Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers,” J. Opt. Soc. Am. A 21(9), 1769 (2004). [CrossRef]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]
- T.-I. Weng and G. Y. Guo, “Band structure of honeycomb photonic crystal slabs,” J. Appl. Phys. 99(9), 093102 (2006). [CrossRef]
- M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89(22), 227402 (2002). [CrossRef] [PubMed]
- P. Lacharmoise, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Optical cavity enhancement of light–sound interaction in acoustic phonon cavities,” Appl. Phys. Lett. 84(17), 3274 (2004). [CrossRef]
- M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (2006). [CrossRef]
- M. Maldovan and E. L. Thomas, “Simultaneous complete elastic and electromagnetic band gaps in periodic structures,” Appl. Phys. B 83(4), 595–600 (2006). [CrossRef]
- S. Sadat-Saleh, S. Benchabane, F. I. Baida, M. P. Bernal, and V. Laude, “Tailoring simultaneous photonic and phononic band gaps,” J. Appl. Phys. 106(7), 074912 (2009). [CrossRef]
- S. Mohammadi, A. A. Eftekhar, and A. Adibi, “Large Simultaneous Band Gaps for Photonic and Phononic Crystal Slabs,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, p. CFY1 (Optical Society of America, 2008). http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2008-CFY1 .
- A. V. Akimov, Y. Tanaka, A. B. Pevtsov, S. F. Kaplan, V. G. Golubev, S. Tamura, D. R. Yakovlev, and M. Bayer, “Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials,” Phys. Rev. Lett. 101(3), 033902 (2008). [CrossRef] [PubMed]
- G. Gantzounis and N. Stefanou, “Layer-multiple-scattering method for photonic crystals of nonspherical particules,” Phys. Rev. B 73(3), 035115 (2006). [CrossRef]
- N. Stefanou, V. Yannopapas, and A. Modinos, “A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132(1-2), 189–196 (2000). [CrossRef]

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