- Email: [email protected]
Engineering of band gap and cavity mode in phononic crystal strip waveguides
Physics Letters A 377 (2013) 2633–2637
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Engineering of band gap and cavity mode in phononic crystal strip waveguides Changsheng Li ∗ , Dan Huang, Jierong Guo, Jianjun Nie Department of Physics and Electronic Sciences, Hunan University of Arts and Science, Changde 415000, PR China
a r t i c l e
i n f o
Article history: Received 9 April 2013 Received in revised form 22 July 2013 Accepted 23 July 2013 Available online 30 July 2013 Communicated by R. Wu
a b s t r a c t The optimal parameters for the largest band gap were investigated in three typical phononic crystal strip waveguides. Single cavity mode was created inside the band gap region by proper design of a defect. The band structures and the displacement distributions were discussed with the variation of the defect. Results show possibilities to guide extremely slow phonon cavity mode in strip waveguide with chosen displacement components, frequencies and symmetries. © 2013 Elsevier B.V. All rights reserved.
Keywords: Phononic crystal Acoustic wave Band gap Cavity mode
1. Introduction Phononic crystal(PC) materials are periodic elastic structures that exhibit a band gap with a certain range of acoustic waves inhibited to propagate [1–3]. The study of acoustic wave in PC structures has attracted attentions since decades ago [4–6]. Phononic band gaps were found in bulk PC materials, multilayer PC materials as well as PC plates, stubbed plates [7–9], and recently PC strips [10–12]. PC structures that exhibit phononic band gap also allow us to localize phonon cavity mode inside the band gap region by introducing a defect [13,14]. With proper design of the defect, the phonon cavity mode can guide a certain range of acoustic waves that we want. It can have many technological applications, such as reduction of noises of a certain range, acoustic filtering and many frequency control devices [15–17]. As the propagation of guided elastic waves along the PC structures can be greatly influenced by the structure, the defect size, and the boundary shapes. The engineering of PC structures and defects becomes increasingly important. Up to now the dominant engineering platform is usually the silicon slab with periodic arrays of sub-micrometer holes or periodic arrays of pillars. Such structures can be fabricated by the widely used silicon-on-insulator technologies. However, for wave guiding or filtering, it is more convenient to have linear structures such as strips. Periodic crystal silicon strips were found to have photonic band gap since decades
*
Corresponding author. Tel.: +86 736 7186121; fax: +86 736 7186122. E-mail address: [email protected] (C. Li).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.07.039
ago and were widely used for confinement of optical waves [18], especially at wavelength 1550 nm in the telecom range. Recent studies found such strip structures can also have phononic band gap and the elastic properties of the strip can greatly influence the optical behavior. Efficient modulation of light pulses through Brillouin scattering by acoustic phonons has been observed at optical frequencies [19]. There has been an emerging research field of the so-called optomechanical or nanomechanical materials [20,21]. As optical strip waveguide were more widely used than its phononic counterpart, right now the focus was mainly on the optical properties of strip and the behavior of optical cavity mode. To enhance the photon–phonon interaction and design new nanomechanical devices, we need to have very high quality phonon cavity mode and study in detail the phononic properties of the cavity. Such as the slowness of the phonon cavity mode, the displacement distributions, and their symmetries. In this Letter we study three strips that can be cut from the typical PC slabs and allow massive production by the widely used silicon-on-insulator technologies. We investigate the band structures of those strips to obtain the largest band gap and search the ideal phonon cavity mode by tuning the defect size. We further design defect that could have phonon cavity mode with extremely slow group velocity compared to the bulk counterpart. The displacement distributions and their symmetries of the cavity mode are also discussed. The rest of the Letter is organized as follows. In Section 2 we describe the simulation model and computing details. Results and discussions of the band structure and cavity mode are presented in Section 3. A short summary is given in Section 4.
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Fig. 1. Schematics of silicon strip waveguides with (a) type i unit cell in shadow, (b) type ii unit cell in shadow, and (c) type iii unit cell in shadow.
Fig. 2. Band structures for silicon strip waveguides type i, type ii, and type iii. Here we choose r /a = 0.3 and h/a = 0.6.
2. Computational details The model is shown in Fig. 1. The so-called strip waveguide is cut from the usual silicon PC plates with vacuum holes. Strip type i and type ii are cut differently from the PC plates with square lattice, while strip type iii is cut from PC plate with triangle lattice. The thickness of the plate is h. The lattice constant is a and the radius is r. The dispersion curve of PC materials can be calculated by various methods, such as plane wave extended method (PWE), finitedifference time-domain (FDTD) method, and finite-element (FE) method. The PWE method is very efficient in simulating band structures of some highly periodic bulk PC materials or PC plates [22]. While FDTD methods are commonly used in simulating dispersion curves and transmission spectrum of some structures with open boundaries. The FE method can be used in most structures and very good at displacement analysis. Results calculated from PWE, FDTD and FE are compared and coincided with each other in previous work [22]. In this work, we employed FE method in band structure calculation. Free stress boundary conditions are applied on all surfaces that contact with vacuum, as elastic waves cannot propagate in vacuum. Periodic conditions were applied on both terminals of the unit cell along the transport direction. Very fine meshes were chosen to converge the results. In calculating the band structure of perfect periodic PC strips, one unit cell is needed in simulation (as shown in the shadow part of Fig. 1). In calculating the band structure of PC strip with defect, super cell calculations were performed (with 6 unit cells on each side the defect) to converge the results. The frequencies are given in the dimensionless unit ωa/2π C t , where ω is frequency, a the length of unit cell in the transport direction, and C t is transverse velocity of elastic waves in silicon (C t = 5844 m/s). In our simulations, the elastic
constants C 11 = 165.7 GPa, C 12 = 63.9 GPa and C 44 = 79.62 GPa and mass density ρ = 2331 kg/m3 . 3. Results and discussion Typical band structures are shown in Fig. 2 for three strip waveguides depicted in Fig. 1. It is impossible to find a phononic band gap for strip type i, even at very big filling factor of vacuum. However, with proper ratios, we can easily observe band gaps in Fig. 2(b) and (c) for strip type ii and type iii. Rough boundaries are necessary for strips to open a phonon gap, which is consistent with previous reports [10]. Besides, high filling factor of vacuum, and a reasonable thickness are essential to open a phonon band gap. The variation of band gap as a function of r /a and h/a is shown in Fig. 3. The grey region is the frequency region of the band gap. From the band map, we observe the optimal ratios for strip type ii (r /a = 0.3 and h/a = 0.8), and type iii (r /a = 0.3 and h/a = 0.6). Now we introduce a defect in PC strip with the aim to localize phonon cavity modes. The defect is created by varying the radius of a single cylinder in the middle of a PC strip waveguide with perfect lattice. The radius of the defect cylinder is defined as r0 . Fig. 4(a) corresponds to the case of type ii perfect strip waveguide where we can define the band gap region when r0 equals to r. Gradually varying the radius of central defect cylinder, the corresponding band structures are shown in Fig. 4(b) to (f). We only observe cavity mode appears in the middle of the gap region at r0 /a = 0.15. Similarly, Fig. 5(a) corresponds to the case of type iii perfect strip waveguide where we define the band gap region as a reference. The band structures of defected PC strips are shown in Fig. 5(b) to (f). Cavity modes are relatively easier to be created in this type. The frequency of cavity modes inside the gap region shifts slowly with the variation of the cavity cylinder. The
C. Li et al. / Physics Letters A 377 (2013) 2633–2637
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Fig. 3. Band maps for silicon strip waveguides type ii (a) as a function of radius for h/a = 0.6 and (b) as a function of height h for r /a = 0.3; For type iii, (c) band map as a function of radius r for h/a = 0.6, (d) as a function of height h for r /a = 0.3.
Fig. 4. Band structures near the band gap region for silicon strip waveguides type ii with varied defect cylinder radius r0 . Here we fixed r /a = 0.35 and h/a = 0.6.
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C. Li et al. / Physics Letters A 377 (2013) 2633–2637
Fig. 5. Band structures near the band gap region for silicon strip waveguides type iii with varied defect cylinder radius r0 . Here we fixed r /a = 0.3 and h/a = 0.6.
Fig. 6. Group velocity of cavity mode versus reduced wave vector (a) for silicon strip waveguides type ii with defect (r0 /a = 0.15), and (b) for silicon strip waveguides type iii with a defect (r0 /a = 0.1).
width of the cavity frequency region can also be controlled by varying the defects, which could be very interesting for acoustic filtering. In order to further investigate the slowness of the cavity mode inside the band gap region, we search the most flat cavity mode in all band structures. We plot the group velocity of the cavity mode in Fig. 6(a) for the single cavity mode (r0 /a = 0.15) in Fig. 4(e). We observe an average group velocity of 0.5 m/s, which is nearly 10,000 decrease compared to the speed of transverse velocity of elastic waves in bulk silicon. In Fig. 6(b) we plot the group velocity for the optimal single cavity mode (r0 /a = 0.1) in Fig. 5(e). Again we see very small group velocity averaged around −2 m/s, which is nearly 2500 times decrease compared to the speed of transverse velocity of elastic waves in bulk silicon. For both type of defected strips, we identify the cavity modes with the slowest group velocity. The spatial displacement distribution
for cavity mode in Fig. 4(e) is displayed in Fig. 7. We observe that the displacement is mainly localized around the defect. The symmetries for each component are shown. We can recognize the symmetries clearly. The spatial distribution of displacement and symmetries for the cavity mode in Fig. 5(e) is displayed in Fig. 8. Around the defect we can see the localizations of each components and their symmetries are quite different compared to Fig. 7. It implies we can filter chosen displacement components with certain frequencies and symmetries by tuning different cavity modes. 4. Summary We have obtained the optimal ratios to achieve the largest band gap in two types of strip waveguides. Ideal cavity mode was de-
C. Li et al. / Physics Letters A 377 (2013) 2633–2637
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signed in the middle of the band gap region. An extremely slow cavity mode is observed with an average group velocity of 0.5 m/s, far less than the speed of bulk waves in silicon. The spatial distributions and the symmetries of the cavity mode are displayed. Our results show possibilities to guide extremely slow phonon cavity mode in strip waveguide with chosen displacement components, frequencies and symmetries. Acknowledgement We acknowledge the support by the National Natural Science Foundation of China under Grant Numbers 11104069 and 61204104. References
Fig. 7. Distribution of displacement of cavity mode at reduced frequency 0.6561 for type ii silicon strip waveguides with defect(r0 /a = 0.15) and r /a = 0.35 and h/a = 0.6 in periodic region.
Fig. 8. Distribution of displacement at reduced frequency 0.43 for type iii silicon strip waveguides with a defect (r0 /a = 0.1). Here we fix r /a = 0.3 and h/a = 0.6 in periodic structures.
[1] M.M. Sigalas, E.N. Economou, J. Sound Vib. 158 (1992) 377. [2] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71 (1993) 2022. [3] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, M.S. Kushwaha, P. Halevi, J. Phys.: Condens. Matter 6 (1994) 8759. [4] M.M. Sigalas, E.N. Economou, J. Appl. Phys. 75 (1994) 2845. [5] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289 (2000) 1734. [6] G. Wang, X. Wen, J. Wen, L. Shao, Phys. Rev. Lett. 93 (2004) 154302. [7] T. Wu, T.T. Wu, J.C. Hsu, Phys. Rev. B 79 (2009) 104306. [8] L. Castro-Arce, B. Manzanares-Martinez, F. Ramos-Mendieta, J. Appl. Phys. 107 (2010) 033515. [9] M. Oudich, Y. Li, B.M. Assouar, Z. Hou, New J. Phys. 12 (2010) 083049. [10] F. Hsu, C. Lee, J. Hsu, T. Huang, C. Wang, P. Chang, Appl. Phys. Lett. 96 (2010) 051902. [11] Y. Yao, F. Wu, X. Zhang, Z. Hou, Phys. Lett. A 376 (2012) 579. [12] N. Aravantinos-Zafiris, M.M. Sigalas, J. Appl. Phys. 111 (2012) 123516. [13] M. Maldovan, E.L. Thomas, Appl. Phys. Lett. 88 (2006) 251907. [14] V. Laude, J.C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J.M. Escalante, A. Martinez, Opt. Express 19 (2011) 144479. [15] J.-C. Hsu, T.-T. Wu, Phys. Rev. B 74 (2006) 144303. [16] S. Mohammadi, A.A. Eftekhar, A. Khelif, W.D. Hunt, A. Adibi, Appl. Phys. Lett. 92 (2008) 221905. [17] M. Eichenfield, J. Chan, R.M. Camacho, K.J. Vahala, O. Painter, Nature, London 462 (2009) 78. [18] J. Joannopoulos, J. Winn, Photonic Crystals: Molding the Flow of Light, Princeton Univ. Press, 2008. [19] A. Akimov, Y. Tanaka, A. Pevtsov, S. Kaplan, V. Golubev, S. Tamura, D. Yakovlev, M. Bayer, Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials, Phys. Rev. Lett. 101 (2008) 033902. [20] A.H. Safavi-Naeini, T.P.M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J.T. Hill, D.E. Chang, O. Painter, Nature 472 (7341) (2011) 69–73. [21] A.H. Safavi-Naeini, J. Chan, J.T. Hill, T.P.M. Alegre, A. Krause, O. Painter, Phys. Rev. Lett. 108 (2012) 033602. [22] Y. El Hassouani, C. Li, Y. Pennec, E.H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, B. Djafari Rouhani, Phys. Rev. B 82 (2010) 155405.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Engineering of band gap and cavity mode in phononic crystal strip waveguides Changsheng Li ∗ , Dan Huang, Jierong Guo, Jianjun Nie Department of Physics and Electronic Sciences, Hunan University of Arts and Science, Changde 415000, PR China
a r t i c l e
i n f o
Article history: Received 9 April 2013 Received in revised form 22 July 2013 Accepted 23 July 2013 Available online 30 July 2013 Communicated by R. Wu
a b s t r a c t The optimal parameters for the largest band gap were investigated in three typical phononic crystal strip waveguides. Single cavity mode was created inside the band gap region by proper design of a defect. The band structures and the displacement distributions were discussed with the variation of the defect. Results show possibilities to guide extremely slow phonon cavity mode in strip waveguide with chosen displacement components, frequencies and symmetries. © 2013 Elsevier B.V. All rights reserved.
Keywords: Phononic crystal Acoustic wave Band gap Cavity mode
1. Introduction Phononic crystal(PC) materials are periodic elastic structures that exhibit a band gap with a certain range of acoustic waves inhibited to propagate [1–3]. The study of acoustic wave in PC structures has attracted attentions since decades ago [4–6]. Phononic band gaps were found in bulk PC materials, multilayer PC materials as well as PC plates, stubbed plates [7–9], and recently PC strips [10–12]. PC structures that exhibit phononic band gap also allow us to localize phonon cavity mode inside the band gap region by introducing a defect [13,14]. With proper design of the defect, the phonon cavity mode can guide a certain range of acoustic waves that we want. It can have many technological applications, such as reduction of noises of a certain range, acoustic filtering and many frequency control devices [15–17]. As the propagation of guided elastic waves along the PC structures can be greatly influenced by the structure, the defect size, and the boundary shapes. The engineering of PC structures and defects becomes increasingly important. Up to now the dominant engineering platform is usually the silicon slab with periodic arrays of sub-micrometer holes or periodic arrays of pillars. Such structures can be fabricated by the widely used silicon-on-insulator technologies. However, for wave guiding or filtering, it is more convenient to have linear structures such as strips. Periodic crystal silicon strips were found to have photonic band gap since decades
*
Corresponding author. Tel.: +86 736 7186121; fax: +86 736 7186122. E-mail address: [email protected] (C. Li).
0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.07.039
ago and were widely used for confinement of optical waves [18], especially at wavelength 1550 nm in the telecom range. Recent studies found such strip structures can also have phononic band gap and the elastic properties of the strip can greatly influence the optical behavior. Efficient modulation of light pulses through Brillouin scattering by acoustic phonons has been observed at optical frequencies [19]. There has been an emerging research field of the so-called optomechanical or nanomechanical materials [20,21]. As optical strip waveguide were more widely used than its phononic counterpart, right now the focus was mainly on the optical properties of strip and the behavior of optical cavity mode. To enhance the photon–phonon interaction and design new nanomechanical devices, we need to have very high quality phonon cavity mode and study in detail the phononic properties of the cavity. Such as the slowness of the phonon cavity mode, the displacement distributions, and their symmetries. In this Letter we study three strips that can be cut from the typical PC slabs and allow massive production by the widely used silicon-on-insulator technologies. We investigate the band structures of those strips to obtain the largest band gap and search the ideal phonon cavity mode by tuning the defect size. We further design defect that could have phonon cavity mode with extremely slow group velocity compared to the bulk counterpart. The displacement distributions and their symmetries of the cavity mode are also discussed. The rest of the Letter is organized as follows. In Section 2 we describe the simulation model and computing details. Results and discussions of the band structure and cavity mode are presented in Section 3. A short summary is given in Section 4.
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Fig. 1. Schematics of silicon strip waveguides with (a) type i unit cell in shadow, (b) type ii unit cell in shadow, and (c) type iii unit cell in shadow.
Fig. 2. Band structures for silicon strip waveguides type i, type ii, and type iii. Here we choose r /a = 0.3 and h/a = 0.6.
2. Computational details The model is shown in Fig. 1. The so-called strip waveguide is cut from the usual silicon PC plates with vacuum holes. Strip type i and type ii are cut differently from the PC plates with square lattice, while strip type iii is cut from PC plate with triangle lattice. The thickness of the plate is h. The lattice constant is a and the radius is r. The dispersion curve of PC materials can be calculated by various methods, such as plane wave extended method (PWE), finitedifference time-domain (FDTD) method, and finite-element (FE) method. The PWE method is very efficient in simulating band structures of some highly periodic bulk PC materials or PC plates [22]. While FDTD methods are commonly used in simulating dispersion curves and transmission spectrum of some structures with open boundaries. The FE method can be used in most structures and very good at displacement analysis. Results calculated from PWE, FDTD and FE are compared and coincided with each other in previous work [22]. In this work, we employed FE method in band structure calculation. Free stress boundary conditions are applied on all surfaces that contact with vacuum, as elastic waves cannot propagate in vacuum. Periodic conditions were applied on both terminals of the unit cell along the transport direction. Very fine meshes were chosen to converge the results. In calculating the band structure of perfect periodic PC strips, one unit cell is needed in simulation (as shown in the shadow part of Fig. 1). In calculating the band structure of PC strip with defect, super cell calculations were performed (with 6 unit cells on each side the defect) to converge the results. The frequencies are given in the dimensionless unit ωa/2π C t , where ω is frequency, a the length of unit cell in the transport direction, and C t is transverse velocity of elastic waves in silicon (C t = 5844 m/s). In our simulations, the elastic
constants C 11 = 165.7 GPa, C 12 = 63.9 GPa and C 44 = 79.62 GPa and mass density ρ = 2331 kg/m3 . 3. Results and discussion Typical band structures are shown in Fig. 2 for three strip waveguides depicted in Fig. 1. It is impossible to find a phononic band gap for strip type i, even at very big filling factor of vacuum. However, with proper ratios, we can easily observe band gaps in Fig. 2(b) and (c) for strip type ii and type iii. Rough boundaries are necessary for strips to open a phonon gap, which is consistent with previous reports [10]. Besides, high filling factor of vacuum, and a reasonable thickness are essential to open a phonon band gap. The variation of band gap as a function of r /a and h/a is shown in Fig. 3. The grey region is the frequency region of the band gap. From the band map, we observe the optimal ratios for strip type ii (r /a = 0.3 and h/a = 0.8), and type iii (r /a = 0.3 and h/a = 0.6). Now we introduce a defect in PC strip with the aim to localize phonon cavity modes. The defect is created by varying the radius of a single cylinder in the middle of a PC strip waveguide with perfect lattice. The radius of the defect cylinder is defined as r0 . Fig. 4(a) corresponds to the case of type ii perfect strip waveguide where we can define the band gap region when r0 equals to r. Gradually varying the radius of central defect cylinder, the corresponding band structures are shown in Fig. 4(b) to (f). We only observe cavity mode appears in the middle of the gap region at r0 /a = 0.15. Similarly, Fig. 5(a) corresponds to the case of type iii perfect strip waveguide where we define the band gap region as a reference. The band structures of defected PC strips are shown in Fig. 5(b) to (f). Cavity modes are relatively easier to be created in this type. The frequency of cavity modes inside the gap region shifts slowly with the variation of the cavity cylinder. The
C. Li et al. / Physics Letters A 377 (2013) 2633–2637
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Fig. 3. Band maps for silicon strip waveguides type ii (a) as a function of radius for h/a = 0.6 and (b) as a function of height h for r /a = 0.3; For type iii, (c) band map as a function of radius r for h/a = 0.6, (d) as a function of height h for r /a = 0.3.
Fig. 4. Band structures near the band gap region for silicon strip waveguides type ii with varied defect cylinder radius r0 . Here we fixed r /a = 0.35 and h/a = 0.6.
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Fig. 5. Band structures near the band gap region for silicon strip waveguides type iii with varied defect cylinder radius r0 . Here we fixed r /a = 0.3 and h/a = 0.6.
Fig. 6. Group velocity of cavity mode versus reduced wave vector (a) for silicon strip waveguides type ii with defect (r0 /a = 0.15), and (b) for silicon strip waveguides type iii with a defect (r0 /a = 0.1).
width of the cavity frequency region can also be controlled by varying the defects, which could be very interesting for acoustic filtering. In order to further investigate the slowness of the cavity mode inside the band gap region, we search the most flat cavity mode in all band structures. We plot the group velocity of the cavity mode in Fig. 6(a) for the single cavity mode (r0 /a = 0.15) in Fig. 4(e). We observe an average group velocity of 0.5 m/s, which is nearly 10,000 decrease compared to the speed of transverse velocity of elastic waves in bulk silicon. In Fig. 6(b) we plot the group velocity for the optimal single cavity mode (r0 /a = 0.1) in Fig. 5(e). Again we see very small group velocity averaged around −2 m/s, which is nearly 2500 times decrease compared to the speed of transverse velocity of elastic waves in bulk silicon. For both type of defected strips, we identify the cavity modes with the slowest group velocity. The spatial displacement distribution
for cavity mode in Fig. 4(e) is displayed in Fig. 7. We observe that the displacement is mainly localized around the defect. The symmetries for each component are shown. We can recognize the symmetries clearly. The spatial distribution of displacement and symmetries for the cavity mode in Fig. 5(e) is displayed in Fig. 8. Around the defect we can see the localizations of each components and their symmetries are quite different compared to Fig. 7. It implies we can filter chosen displacement components with certain frequencies and symmetries by tuning different cavity modes. 4. Summary We have obtained the optimal ratios to achieve the largest band gap in two types of strip waveguides. Ideal cavity mode was de-
C. Li et al. / Physics Letters A 377 (2013) 2633–2637
2637
signed in the middle of the band gap region. An extremely slow cavity mode is observed with an average group velocity of 0.5 m/s, far less than the speed of bulk waves in silicon. The spatial distributions and the symmetries of the cavity mode are displayed. Our results show possibilities to guide extremely slow phonon cavity mode in strip waveguide with chosen displacement components, frequencies and symmetries. Acknowledgement We acknowledge the support by the National Natural Science Foundation of China under Grant Numbers 11104069 and 61204104. References
Fig. 7. Distribution of displacement of cavity mode at reduced frequency 0.6561 for type ii silicon strip waveguides with defect(r0 /a = 0.15) and r /a = 0.35 and h/a = 0.6 in periodic region.
Fig. 8. Distribution of displacement at reduced frequency 0.43 for type iii silicon strip waveguides with a defect (r0 /a = 0.1). Here we fix r /a = 0.3 and h/a = 0.6 in periodic structures.
[1] M.M. Sigalas, E.N. Economou, J. Sound Vib. 158 (1992) 377. [2] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71 (1993) 2022. [3] J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, M.S. Kushwaha, P. Halevi, J. Phys.: Condens. Matter 6 (1994) 8759. [4] M.M. Sigalas, E.N. Economou, J. Appl. Phys. 75 (1994) 2845. [5] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289 (2000) 1734. [6] G. Wang, X. Wen, J. Wen, L. Shao, Phys. Rev. Lett. 93 (2004) 154302. [7] T. Wu, T.T. Wu, J.C. Hsu, Phys. Rev. B 79 (2009) 104306. [8] L. Castro-Arce, B. Manzanares-Martinez, F. Ramos-Mendieta, J. Appl. Phys. 107 (2010) 033515. [9] M. Oudich, Y. Li, B.M. Assouar, Z. Hou, New J. Phys. 12 (2010) 083049. [10] F. Hsu, C. Lee, J. Hsu, T. Huang, C. Wang, P. Chang, Appl. Phys. Lett. 96 (2010) 051902. [11] Y. Yao, F. Wu, X. Zhang, Z. Hou, Phys. Lett. A 376 (2012) 579. [12] N. Aravantinos-Zafiris, M.M. Sigalas, J. Appl. Phys. 111 (2012) 123516. [13] M. Maldovan, E.L. Thomas, Appl. Phys. Lett. 88 (2006) 251907. [14] V. Laude, J.C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J.M. Escalante, A. Martinez, Opt. Express 19 (2011) 144479. [15] J.-C. Hsu, T.-T. Wu, Phys. Rev. B 74 (2006) 144303. [16] S. Mohammadi, A.A. Eftekhar, A. Khelif, W.D. Hunt, A. Adibi, Appl. Phys. Lett. 92 (2008) 221905. [17] M. Eichenfield, J. Chan, R.M. Camacho, K.J. Vahala, O. Painter, Nature, London 462 (2009) 78. [18] J. Joannopoulos, J. Winn, Photonic Crystals: Molding the Flow of Light, Princeton Univ. Press, 2008. [19] A. Akimov, Y. Tanaka, A. Pevtsov, S. Kaplan, V. Golubev, S. Tamura, D. Yakovlev, M. Bayer, Hypersonic modulation of light in three-dimensional photonic and phononic band-gap materials, Phys. Rev. Lett. 101 (2008) 033902. [20] A.H. Safavi-Naeini, T.P.M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J.T. Hill, D.E. Chang, O. Painter, Nature 472 (7341) (2011) 69–73. [21] A.H. Safavi-Naeini, J. Chan, J.T. Hill, T.P.M. Alegre, A. Krause, O. Painter, Phys. Rev. Lett. 108 (2012) 033602. [22] Y. El Hassouani, C. Li, Y. Pennec, E.H. El Boudouti, H. Larabi, A. Akjouj, O. Bou Matar, V. Laude, N. Papanikolaou, A. Martinez, B. Djafari Rouhani, Phys. Rev. B 82 (2010) 155405.