聲子晶體是由兩種或兩種以上不同的彈性材料或流體週期排列而形成的結構，此人造結構擁有聲子能隙的特殊現象，可隔絕聲波或彈性波在聲子晶體上的傳遞，使特定角度與頻率入射的聲波無法傳遞通過聲子晶體，可應用於濾波器或隔音裝置上。破壞完美聲子晶體的週期排列，將會在能隙範圍產生缺陷模態，在缺陷模態的頻率之下能夠將聲波或彈性波侷限在聲子晶體的缺陷之中，缺陷模態所對應的頻帶即為缺陷頻帶。移除一聲子晶體填充物形成聲子晶體的點缺陷，在其頻散曲線上會在能隙範圍內出現缺陷頻帶，此缺陷頻帶可視為穿透能帶，缺陷頻帶上的頻率可通過此含點缺陷聲子晶體；點缺陷亦可作為聲子晶體共振腔，共振腔的共振頻率即為缺陷頻帶的頻率。當入射聲波之頻率為共振頻率時，聲波將被侷限在共振腔之中，可達到濾波的效果，且共振腔中的聲波壓力將被大幅提升。
本文使用平面波展開法與超晶胞法計算含點缺陷聲子晶體之頻散曲線，並利用有限元素軟體計算週期結構之穿透頻譜與聲子晶體共振腔中的壓力，移除數個填充物將缺陷放大，並在周圍以鋼柱排列成圓形，分析在共振腔中產生的迴廊模態共振，討論共振腔的品質因子與聲壓放大特性、是否可產生環形共振及在共振腔中央引入待測物是否會影響品質因子及共振頻率，並在共振腔旁引入波導增加結構的整合性。以往迴廊模態多數被用在光學應用上，今應用在聲學研究上，可達到比點缺陷共振腔更高的品質因子和更大的聲壓增強，將可達到更精確的濾波效果。

Sonic crystals (phononic crystals) are periodic elastic composite materials. Such artificial crystals can exhibit acoustic or elastic band gaps in which sound and vibration are all forbidden in any direction, giving rise to prospective applications such as elastic/acoustic filters and noise/vibration isolations. One particularly interesting aspect of sonic crystals is the possibility of creating crystal defects to confine the elastic/acoustic waves in localized modes. Because of locally breaking the periodicity of the structure, the defect modes can be created within the band gaps, which are strongly localized around the local defect. The point defect is created by removing a single rod from the middle of the perfect periodic structure. There exist the defect bands in the absoulate band gap. The acoustic wave can propagate through the sonic crystal, since the defect band acts as a pass band in the band gap. The point defect can also act as the resonant cavity. At the frequency of the defect band, which is the resonant frequency, the acoustic waves should be localized in the resonant cavity and the pressures in the cavity are enhanced.
Sonic WGM has not been analyzed in phononic crystal previously. In this paper, we study the WGM which formed by removing several cylinders away from phononic crystal with PWE method and analyze the pressure in the defect to confirm acoustic WGM resonance. The advantages of the of WGM resonance in this large-size cavity are the pressure and sensitivity are higher than the single defect resonance . Considering if we want to use the device as the sensor, it is important to insert a waveguide besides the cavity. So we also inserted a waveguide and investigate coupling between cavity and waveguide.

[1] E. Yablonovitch, “Inhibited spontaneous emission in slid-state physics and electronics”, Phys. Rev. Lett. Vol. 58, No. 20, pp. 2059 (1987)
[2] S. John, “Strong localization of photons in certain disordered dielectric superlattice”, Phys. Rev. Lett. Vol. 58, No. 23, pp. 2486 (1987)
[3] L. Brillouin, “Wave propagation in periodic structure”, 2nded. Dover, New York (1953)
[4] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B.Djafari-Rouhani, ‘Acoustic band structure of periodic elastic composites’, Phys. Rev. Lett. Vol. 71, No. 13, 2022 (1993)
[5] M. S. Kushwaha, P. Halevi, ‘Band-gap engineering in periodic elastic composites’, Appl. Phys. Lett. Vol. 64, No.9, 1805 (1994)
[6] M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski, and B.Djafari-Rouhani, ‘Theory of acoustic band structure of periodic elastic composites’, Phys. Rev. B. Vol. 49, No. 4, 2313 (1994)
[7] M. S. Kushwaha, ‘Classical band structure of periodic elastic composites’, Internationa1 Journal of Modern Physics B Vol. 10, No. 9, 977 (1996)
[8] M. S. Kushwaha, P. Halevi, ‘Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders’, Appl. Phys. Lett. Vol.69, No. 1, 31 (1996)
[9] C. Goffaux and J. P. Vigneron, ‘Theoretical study of a tunable phononic band gap system’, Phys. Rev. B, Vol. 64, No. 7, 075118 (2001)
[10] J. O. Vasseur, P. A. Deymier, A. Khelif, B. Djafrari-Rouhani, A. Akjouj, L. Dobrzynski, N. Fettouhi, and J. Zemmouri, ‘Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: A theoretical and experimental study’, Phys. Rev. E Vol.65, No. 5, 056608 (2002)
[11] A. Khelif, P. A. Deymier, B. Djafari-Rouhani, J. O. Vasseur, and L. Dobrzynski, ‘Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency’, J. Appl. Phys. Vol. 94, No.3, 1308 (2003)
[12] Y. Pennec, B. Djafari-Rouhani, J. O. Vasseur, A. Khelif, and P. A. Deymier, ‘Tunable filtering and demultiplexing in phononic crystals with hollow cylinders’, Phys. Rev. E Vol. 69, No.4, 046608 (2004)
[13] W. Kuang, Z. Hou, and Y. Liu, ‘The effects of shapes and symmetries of scatterers on the phononic band gap in 2D phononic crystals’, Phys. Let. A Vol. 332, No. 6, 481 (2004)
[14] X. Zhang, Y. Liu, F. Wu, and Z. Liu, ‘Large two-dimensional band gaps in three-component phononic crystals’, Phys. Let. A Vol. 317, No. 2 144 (2003)
[15] M. Kafesaki, and E. N. Economou, ‘Multiple-scattering theory for three-dimensional periodic acoustic composites’, Phys. Rev. B Vol. 60, No. 17, 11993(1999)
[16] Z. Hou, F. Wu, and Y. Liu, ‘Phononic crystals containing piezoelectric material’, Solid State Communications Vol. 130 , 745 (2004)
[17] T. T. Wu, Z. G. Huang, and S. Lin, ‘Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy’, Phys. Rev. B Vol. 69, No. 9, 094301 (2004)
[18] D. Torrent and J. Sánchez-Dehesa, ‘Radial wave crystals : Radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves’, Phys. Rew. Lett. Vol. 103, No. 6, pp. 064301 (2008)
[19] D. Torrent and J. Sánchez-Dehesa, “Acoustic resonances in two-dimensional radial sonic crystal shells”, New J. Phys. Vol. 12, No. 7, pp. 073034 (2010)
[20] M. M. Sigalas, “Elastic wave band gaps and defect states in two-dimensional composites”, J. Acoust. Soc. Am. Vol. 101, No. 3, pp. 1256 (1996)
[21] M. M. Sigalas, “Defect states of acoustic waves in a two-dimensional lattice of solid cylinders”, J. Appl. Phys. Vol. 84, No. 6, pp. 3026 (1998)
[22] F. Wu, Z. Hou, Z. Liu and Y. Liu, “Point defect states in two-dimensional phononic crystals”, Phys. Lett. A Vol. 292, No. 3, pp. 198 (2001)
[23] F. Wu, Z. Liu, Y. Liu, “Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals”, Phys. Rev. E Vol. 69, No. 6, pp. 066609 (2004)
[24] J. Chen, J. C. Cheng and B. Li, “Dynamics of elastic waves in two-dimensional phononic crystals with chaotic defect”, Appl. Phys. Lett. Vol. 91, No. 12, pp. 121902 (2007)
[25] L. Y. Wu, L. W. Chen and C. M. Liu, “Acoustic pressure in cavity of variously sized two-dimensional sonic crystals with various filling fractions”, Phys. Lett. A Vol. 373, No. 12-13, pp. 1189 (2009)
[26] L. Y. Wu, L. W. Chen and C. M. Liu, “Experimental investigation of the acoustic pressure in cavity of a two-dimensional sonic crystal”, Physica B Vol. 404, No. 12-13, pp. 1766 (2009)
[27] L. Y. Wu, L. W. Chen and C. M. Liu, “Acoustic energy harvesting using resonant cavity of a sonic crystal”, Appl. Phys. Lett. Vol. 95, No. 1, pp. 013506 (2009)
[28] L. Y. Wu and L. W. Chen, “Wave propergation in a 2D sonic crystal with a Helmholtz resonant defect”, J. Phys. D : Appl. Phys. Vol. 43, No. 5, pp. 055401 (2010)
[29] X. Zhang, H. Dan, F. Wu and Z. Liu, “Point defect states in 2D acoustic band gap materials consisting of solid cylinders in viscous liquid”, J. Phys. D : Appl. Phys. Vol. 41, No. 15, pp. 155110 (2008)
[30] http://web2.ctsh.hcc.edu.tw/~s9311104/u4.html

[31] A.N.Oraevsky “whispering-gallery waves＂Quantum Electronics,Vol.32,No.5,pp.377 2002
[32] Han-Youl Ryu ,Masaya Notomi and Yong-Hee Lee “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities” Applied Physical Letters VOL.83, No. 21(2003)
[33] Han-Youl Ryu ,Masaya Notomi , Guk-HyunKim and Yong-Hee Lee “ High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity” Optics Express 1708 Vol. 12, No. 8 (2004)
[34] Mingxin Xing, Wanhua Zheng, Yejin Zhang, Gang Ren, Xiaoyu Du, Ke Wang, Lianghui Chen “The whispering gallery mode in photonic crystal ring cavity” SPIE Vol. 6984, 698438, (2008)
[35] Po-Tsung Lee, Tsan-Wen Lu, Chia-Ming Yu, and Chung-Chuan Tseng “Photonic crystal circular-shaped microcavity and its uniform cavity-waveguide coupling property due to presence of whispering gallery mode” Optics Express Vol. 15, No. 15 (2007)
[36] Ken’ichi Nagahara, Masato Morifuji , Masahiko Kondow” Optical coupling between a cavity mode and a waveguide in a two-dimensional photonic crystal” Science Direct Vol. 9 261–268 (2011)
[37] M. S. Kushwaha, P. Halevi, L. Dobrzynski and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites”, Phys. Rev. Lett. Vol. 71, No. 13, pp.2022 (1993)
[38] M. S. Kushwaha, “Classical band structure of periodic elastic composites”, Int. J. Mod. Phys. B Vol. 10, No. 9, pp.977 (1996)
[39] M. S. Kushwaha and P. Halevi, “Band-gap engineering in periodic elastic composites”, Appl. Phys. Lett. Vol. 64, No. 9, pp.1085 (1994)
[40] M. S. Kushwaha, P. Halevi, G. Martínez, L. Dobrzynski and B. Djafari-Rouhani, “Theory of acoustic band structure of periodic elastic composites”, Phys. Rev. B Vol. 49, No. 4, pp.2313 (1994)
[41] M. S. Kushwaha and P. Halevi, “Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders”, Appl. Phys. Lett. Vol. 69, No. 1, pp.31 (1996)
[42] M. S. Kushwaha, “Stop-bands for periodic metallic rods: Sculptures that canfilter the noice”, Appl. Phys. Lett. Vol. 70, No. 24, pp.3218 (1997)
[43] M. S. Kushwaha and B. Djafari-Rouhani, “Giant sonic band in two-dimensional periodic system of fluids”, J. Appl. Phys. Vol. 84, No. 9, pp.4677 (1998)
[44] COMSOL 3.5a The COMSOL Group, Stockholm, Sweden (2009)
[45] J. N. Reddy, An introduction to the finite element method 3rded. McGraw-Hill, New York (2006)