## Three-dimensional dielectric phoxonic crystals with network topology |

Optics Express, Vol. 21, Issue 3, pp. 2727-2732 (2013)

http://dx.doi.org/10.1364/OE.21.002727

Acrobat PDF (1201 KB)

### Abstract

We theoretically demonstrate the existence of simultaneous large complete photonic and phononic bandgaps in three-dimensional dielectric phoxonic crystals with a simple cubic lattice. These phoxonic crystals consist of dielectric spheres on the cubic lattice sites connected by thin dielectric cylinders. The simultaneous photonic and phononic bandgaps can exist over a wide range of geometry parameters. The vibration modes corresponding to the phononic bandgap edges are the local torsional resonances of the dielectric spheres and rods. Detailed discussion is presented on the variation of the photonic and phononic bandgaps with the geometry of the structure. Optimal geometry which generates large phoxonic bandgaps is suggested.

© 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]

3. 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]

4. T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. **94**(22), 223902 (2005). [CrossRef] [PubMed]

5. M. Eichenfield, R. M. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature **459**(7246), 550–555 (2009). [CrossRef] [PubMed]

6. 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]

7. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature **462**(7269), 78–82 (2009). [CrossRef] [PubMed]

8. I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B **82**(17), 174303 (2010). [CrossRef]

9. N. Papanikolaou, I. E. Psarobas, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Light modulation in phoxonic nanocavities,” Microelectron. Eng. **90**, 155–158 (2012). [CrossRef]

6. 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]

10. 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]

12. D. Bria, M. B. Assouar, M. Oudich, Y. Pennec, J. Vasseur, and B. Djafari-Rouhani, “Opening of simultaneous photonic and phononic band gap in two-dimensional square lattice periodic structure,” J. Appl. Phys. **109**(1), 014507 (2011). [CrossRef]

13. 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]

14. N. Papanikolaou, I. E. Psarobas, and N. Stefanou, “Absolute spectral gaps for infrared light and hypersound in three-dimensional metallodielectric phoxonic crystals,” Appl. Phys. Lett. **96**(23), 231917 (2010). [CrossRef]

15. Y. Pennec, B. D. Rouhani, C. Li, J. M. Escalante, A. Martinez, S. Benchabane, V. Laude, and N. Papanikolaou, “Band gaps and cavity modes in dual phononic and photonic strip waveguides,” AIP Adv. **1**(4), 041901 (2011). [CrossRef]

16. F. L. Hsiao, C. Y. Hsieh, H. Y. Hsieh, and C. C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. **100**(17), 171103 (2012). [CrossRef]

17. S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express **18**(9), 9164–9172 (2010). [CrossRef] [PubMed]

18. Y. Pennec, B. D. 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**(13), 14301–14310 (2010). [CrossRef] [PubMed]

19. Y. El Hassouani, C. Li, Y. Pennec, E. H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, and B. Djafari Rouhani, “Dual phononic and photonic band gaps in a periodic array of pillars deposited on a thin plate,” Phys. Rev. B **82**(15), 155405 (2010). [CrossRef]

20. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express **18**(14), 14926–14943 (2010). [CrossRef] [PubMed]

## 2. Design, methods of calculation and band structures

*a*, sphere radius

*R*and cylinder radius

*r*. PXCs in a SC lattice can be easily and economically fabricated because of the inherent simplicity of the geometry [21

21. H. S. Sözüer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B **10**(2), 296–302 (1993). [CrossRef]

23. M. Maldovan and E. L. Thomas, “Photonic crystals: six connected dielectric networks with simple cubic symmetry,” J. Opt. Soc. Am. B **22**(2), 466–473 (2005). [CrossRef]

22. R. Biswas, M. M. Sigalas, K. M. Ho, and S. Y. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B **65**(20), 205121 (2002). [CrossRef]

23. M. Maldovan and E. L. Thomas, “Photonic crystals: six connected dielectric networks with simple cubic symmetry,” J. Opt. Soc. Am. B **22**(2), 466–473 (2005). [CrossRef]

24. COMSOL Group, “COMSOL Multiphysics,” http://www.comsol.com/

*n*= 3.6, mass density

*ρ*= 2330 kg/m

^{3}, transverse and longitudinal wave velocities

*c*= 5360 m/s and

_{t}*c*= 8950 m/s, respectively [10

_{l}10. 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]

*r*= 0.1

*a*and

*R*= 0.375

*a*. Figures 1(a) and 1(b) present the phononic and photonic band structures which are normalized by 2π

*c*/

*a*and 2π

*c*/

_{t}*a*, respectively, with

*c*being the velocity of light in vacuum and

*c*the transverse wave velocity of the dielectric matrix. The corresponding phononic and photonic bandgaps are of the normalized frequency ranges 0.346 <

_{t}*ωa*/2π

*c*< 0.827 (with the gap-to-midgap ratio of 82%) and 0.406 <

_{t }*ωa*/2π

*c*< 0.455 (with the gap-to-midgap ratio of 11.4%), respectively. This particular structure and corresponding bandgaps are dimensionless. In view of telecommunication applications, the PTBG wavelength

*λ*is close to 1550 nm. With this value as the midgap frequency, the lattice constant is

*a*= 667 nm, the sphere and cylinder radii are

*R*= 250 nm and

*r*= 67 nm, respectively. In turn, the midgap frequency of the PNBG is

*f*= 4.7 GHz. This particular example shows that the designed structure indeed exhibits dual bandgaps.

## 3. Detailed numerical results and discussions

*r*/

*a*for the fixed sphere radius

*R*= 0.35

*a*. As shown in Fig. 2(a), the maximum PNBG appears with the minimum cylinder radius, and the bandgap widths of two bandgaps (between the 6th and 7th, 12th and 13th bands) generally decrease as the cylinder radius

*r*increases. The upper and lower bandgaps close at

*r*= 0.095

*a*and

*r*= 0.18

*a*, respectively. One can see that the passing band (between 7th and 12th bands) is compressed as

*r*decreases, and ultimately covers a quite narrow frequency range of 0.51 <

*ωa*/2π

*c*< 0.52 at

_{t }*r*= 0.02

*a*. For the PTBG map shown in Fig. 2(b), the gap opens up at

*r*= 0.07

*a*, and widens up as

*r*increases until

*r*= 0.1

*a*, and then becomes narrower with

*r*increasing. Both the upper and lower bandgap edges decrease with

*r*increasing. The PTBG exists when 0.07 <

*r*/

*a*< 0.2. It can be seen that dual bandgaps appear in the normalized cylinder radius range of 0.07 <

*r*/

*a*< 0.18. The optimal choice of the cylinder radius is

*r*= 0.1

*a*in which case the phononic and photonic bandgaps are over the normalized frequency ranges 0.349 <

*ωa*/2π

*c*< 0.763 and 0.418 <

_{t }*ωa*/2π

*c*< 0.478, respectively.

*r*= 0.1

*a*, the variations of the PNBG and PTBG as functions of

*R*/

*a*are shown in Fig. 3 . We can observe that there are three PNBGs between the 6th and 7th, 12th and 13th, 15th and 16th bands in Fig. 3(a). Unfortunately only the first bandgap between the 6th and 7th bands is wide, the others are quite narrow and less applicable. So in what follows we only consider the widest one. The PNBG opens up at approximately

*R*= 0.225

*a*at the normalized frequency of 0.4204, and widens up as the sphere radius

*R*increases. The PNBG extends to maximum at

*R*= 0.4

*a*with the gap-to-midgap ratio of 84.5%, and then becomes narrower slightly as

*R*increases. Figure 3(b) shows that the PTBG appears in the similar values of

*R*/

*a*, and the widest gap is found at

*R*= 0.35

*a*. The dual bandgaps appear in a wide normalized sphere radius range of 0.24 <

*R*/

*a*< 0.45.

*R*/

*a*and

*r*/

*a*, variations of the gap-to-midgap ratio of the first bandgap for the phononic and photonic cases with both geometry parameters

*R*/

*a*and

*r*/

*a*are shown in Figs. 4(a) and 4(b), respectively. For the PNBG map, the gap-to-midgap ratio decreases with

*r*/

*a*increasing. For a given

*r*/

*a*, this bandgap is widest at an intermediate value of

*R*/

*a*and becomes narrower as

*R*/

*a*deviates from this value. The variation of the PTBG with

*R*/

*a*and

*r*/

*a*is more complex. There is a region of the geometry parameters (near

*R*= 0.35

*a*and

*r*= 0.105

*a*) for the widest PTBG; and the gap-to-midgap ratio decreases as the combination of

*R*/

*a*and

*r*/

*a*deviates from these values, see Fig. 4(b). One can see that the simultaneous PNBG and PTBG exist in most area of the geometry parameters in Fig. 4. And in the range of 0.365 <

*R*/

*a*< 0.41 and 0.1 <

*r*/

*a*< 0.125, the gap-to-midgap ratios for the phononic and photonic bandgaps exceed 50% and 8%, respectively. So it is flexible to design the detailed structures according to the applications and fabrication difficulties.

22. R. Biswas, M. M. Sigalas, K. M. Ho, and S. Y. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B **65**(20), 205121 (2002). [CrossRef]

## 4. Conclusion

25. G. Gantzounis, N. Papanikolaou, and N. Stefanou, “Nonlinear interactions between high-Q optical and acoustic modes in dielectric particles,” Phys. Rev. B **84**(10), 104303 (2011). [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. | M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett. |

4. | T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. |

5. | M. Eichenfield, R. M. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature |

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

7. | M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature |

8. | I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B |

9. | N. Papanikolaou, I. E. Psarobas, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Light modulation in phoxonic nanocavities,” Microelectron. Eng. |

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

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

12. | D. Bria, M. B. Assouar, M. Oudich, Y. Pennec, J. Vasseur, and B. Djafari-Rouhani, “Opening of simultaneous photonic and phononic band gap in two-dimensional square lattice periodic structure,” J. Appl. Phys. |

13. | 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. |

14. | N. Papanikolaou, I. E. Psarobas, and N. Stefanou, “Absolute spectral gaps for infrared light and hypersound in three-dimensional metallodielectric phoxonic crystals,” Appl. Phys. Lett. |

15. | Y. Pennec, B. D. Rouhani, C. Li, J. M. Escalante, A. Martinez, S. Benchabane, V. Laude, and N. Papanikolaou, “Band gaps and cavity modes in dual phononic and photonic strip waveguides,” AIP Adv. |

16. | F. L. Hsiao, C. Y. Hsieh, H. Y. Hsieh, and C. C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. |

17. | S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express |

18. | Y. Pennec, B. D. 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 |

19. | Y. El Hassouani, C. Li, Y. Pennec, E. H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, and B. Djafari Rouhani, “Dual phononic and photonic band gaps in a periodic array of pillars deposited on a thin plate,” Phys. Rev. B |

20. | A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express |

21. | H. S. Sözüer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B |

22. | R. Biswas, M. M. Sigalas, K. M. Ho, and S. Y. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B |

23. | M. Maldovan and E. L. Thomas, “Photonic crystals: six connected dielectric networks with simple cubic symmetry,” J. Opt. Soc. Am. B |

24. | COMSOL Group, “COMSOL Multiphysics,” http://www.comsol.com/ |

25. | G. Gantzounis, N. Papanikolaou, and N. Stefanou, “Nonlinear interactions between high-Q optical and acoustic modes in dielectric particles,” Phys. Rev. B |

**OCIS Codes**

(160.1050) Materials : Acousto-optical materials

(220.4880) Optical design and fabrication : Optomechanics

(350.7420) Other areas of optics : Waves

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 18, 2012

Revised Manuscript: January 16, 2013

Manuscript Accepted: January 18, 2013

Published: January 28, 2013

**Citation**

Tian-Xue Ma, Yue-Sheng Wang, Yan-Feng Wang, and Xiao-Xing Su, "Three-dimensional dielectric phoxonic crystals with network topology," Opt. Express **21**, 2727-2732 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2727

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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]
- 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]
- T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett.94(22), 223902 (2005). [CrossRef] [PubMed]
- M. Eichenfield, R. M. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459(7246), 550–555 (2009). [CrossRef] [PubMed]
- 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]
- M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462(7269), 78–82 (2009). [CrossRef] [PubMed]
- I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B82(17), 174303 (2010). [CrossRef]
- N. Papanikolaou, I. E. Psarobas, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Light modulation in phoxonic nanocavities,” Microelectron. Eng.90, 155–158 (2012). [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. B83(4), 595–600 (2006). [CrossRef]
- D. Bria, M. B. Assouar, M. Oudich, Y. Pennec, J. Vasseur, and B. Djafari-Rouhani, “Opening of simultaneous photonic and phononic band gap in two-dimensional square lattice periodic structure,” J. Appl. Phys.109(1), 014507 (2011). [CrossRef]
- 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]
- N. Papanikolaou, I. E. Psarobas, and N. Stefanou, “Absolute spectral gaps for infrared light and hypersound in three-dimensional metallodielectric phoxonic crystals,” Appl. Phys. Lett.96(23), 231917 (2010). [CrossRef]
- Y. Pennec, B. D. Rouhani, C. Li, J. M. Escalante, A. Martinez, S. Benchabane, V. Laude, and N. Papanikolaou, “Band gaps and cavity modes in dual phononic and photonic strip waveguides,” AIP Adv.1(4), 041901 (2011). [CrossRef]
- F. L. Hsiao, C. Y. Hsieh, H. Y. Hsieh, and C. C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett.100(17), 171103 (2012). [CrossRef]
- S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express18(9), 9164–9172 (2010). [CrossRef] [PubMed]
- Y. Pennec, B. D. 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. Express18(13), 14301–14310 (2010). [CrossRef] [PubMed]
- Y. El Hassouani, C. Li, Y. Pennec, E. H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, and B. Djafari Rouhani, “Dual phononic and photonic band gaps in a periodic array of pillars deposited on a thin plate,” Phys. Rev. B82(15), 155405 (2010). [CrossRef]
- A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express18(14), 14926–14943 (2010). [CrossRef] [PubMed]
- H. S. Sözüer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B10(2), 296–302 (1993). [CrossRef]
- R. Biswas, M. M. Sigalas, K. M. Ho, and S. Y. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B65(20), 205121 (2002). [CrossRef]
- M. Maldovan and E. L. Thomas, “Photonic crystals: six connected dielectric networks with simple cubic symmetry,” J. Opt. Soc. Am. B22(2), 466–473 (2005). [CrossRef]
- COMSOL Group, “COMSOL Multiphysics,” http://www.comsol.com/
- G. Gantzounis, N. Papanikolaou, and N. Stefanou, “Nonlinear interactions between high-Q optical and acoustic modes in dielectric particles,” Phys. Rev. B84(10), 104303 (2011). [CrossRef]

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