橢圓柱聲子晶體漸變折射率聚焦與頻散關係之研究

類比於光子晶體，聲子晶體是週期性排列的散射子在基底中的結構。首先本文研究了二維圓柱及橢圓柱聲子晶體的頻散關係，包括帶隙、部分帶隙、以及定向傳播，然後是漸變折射率聲子晶體。利用平面波展開法得到頻帶結構，經由等頻曲線分析其波傳特性後，與有限元素法模擬做比較。在不同的頻率下透過不同填充率或不同材料的聲子晶體下所對應的折射率都會不同，藉由光學漸變折射率的概念，設計了二維漸變折射率聲子晶體去操縱聲波的行進方向。在這裡漸變折射率聲子晶體是由鋼柱週期排列成正方晶格，背景為空氣。透過平面波展開法來計算折射率，以及有限元素法來模擬波傳遞的現象。在二維聲子晶體徑向上透過改變不同旋轉角的橢圓柱，形成不同的折射率，用雙曲正割函數來設計折射率分佈，最後討論經由漸變折射率聲子晶體透鏡控制聲波聚焦的結果，包括不同層數及不同漸變係數的影響。以上二維聲子晶體波傳理論可進而發展至三維聲子晶體以及應用在未來的聲學元件與超音波儀器上。

關鍵字

聲子晶體；漸變折射率；聚焦

並列摘要

Similar to photonic crystals, sonic crystals (SC) is a periodic array composed of scatters embedded in a host material. Here we study the dispersion characteristics of 2D sonic crystal with circular and elliptic rods first, including band gap, partial band gap and directional propagation, and then the gradient-index sonic crystals. The use of plane wave expansion method is to obtain the band structure. We compare the propagation characteristics analyzed by the equifrequecy surfaces with the simulation results by finite element method. The refractive indexes for various filling fractions, materials or frequencies of a sonic crystals are all different. By the concept of gradient-index in optics, we designed a two-dimensional gradient-index (GRIN) sonic crystals to control the propagations of acoustic waves. The GRIN SC was composed of square array steel cylinders in the background of air. The refractive index is obtained by plane wave expansion method, and the wave propagation is simulated by finite-element method. Altering the rotation angles of the elliptic rods in 2D SC to make various refractive indexes and the refractive index distribution is designed as hyperbolic secant function. The results and discussions for different layers and gradient coefficient demonstrate the GRIN SC lens focusing behavior of acoustic wave. It is anticipated that this study may be applied in 3D acoustic assembled devices or in ultrasonic devices.

並列關鍵字

Sonic crystals ； Gradient index ； Focusing

參考文獻

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[3] S. Tamura, D. C. Hurley, and J. P. Wolfe, "Acoustic-phonon propagation in superlattices," Phys. Rev. Rev. B 38(2), 1427-1449 (1998).

[4] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, and B. Djafari-Rouhani, "Theory of acoustic band structure of periodic elastic composites," Phys. Rev. Rev. B 49(4),2313-2322 (1994).

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國際替代計量

橢圓柱聲子晶體漸變折射率聚焦與頻散關係之研究

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