板波於聲子晶體平板聚焦之研究

Abstract

聲子晶體(Phononic Crystals)是一種週期性的複合材料，其結構為均質填充物週期性地排列於具有不同物理性質固體或液體中。主要是利用不同材料其彈性矩陣和質量密度等物理性質上的差異使得聲波在聲子晶體中傳播會造成特殊之波傳行為，如頻溝現象(band gap)、負折射現象(negative refraction)、…等。 本文主要為利用有限元素法(finite element method, FEM)來分析板波(Lamb waves)在聲子晶體平板結構中聚焦之行為，並進行實驗驗證。文中主要探討兩種聲子晶體平板所造成不同機制的聚焦行為，文中第一部分「漸變型聲子晶體平板」為探討改變單位晶胞(unit cell)的填充率進而產生聚焦的波傳行為。藉由填充率與折射指數之間的關係，使折射指數符合光學上的漸變指數(Gradient-Index Optics)，即為使隨洞徑變化的折射指數分布符合hyper secant曲線進而設計漸變洞徑尺寸大小。實驗方面包含了毫米級鋁板漸變型聲子晶體以及微米級矽基版漸變型聲子晶體之聚焦實驗，成果顯示與數值預期結果相當一致，相較於波源振幅，聚焦後之振幅成功且有效地放大接近兩倍。 文中第二部分為「負折射型聲子晶體」，主要為探討三角晶格排列之聲子晶體平板所產生之負折射現象，利用此現象加以設計使點波源激發後經過聲子晶體平板產生聚焦行為，並藉由等頻面(equi-frequency surface, EFS)和施奈爾定律(Snell–Descartes laws)來分析及預測入射角度和折射角度，接著藉由實驗加以驗證。而負折射實驗部分為採用雷射超音波之點波源激發，利用探頭接收訊號，再藉由訊號處理得到不同頻率下的波傳行為再加以與模擬結果比對。實驗結果與模擬預期相當吻合，成功在一個區域同時聚焦40-45 kHz的反對稱模態，而相較於漸變型聲子晶體，雖然沒有放大振幅的效果，但能提升解析度。

Phononic crystals (PC) are composite materials which consist of homogeneous elastic inclusions distributed periodically in a background medium characterized by different physical properties, such as mass density and elastic stiffness. Due to the special scattering effect in such a structure, there are many special properties existed. In this study, we investigate two different mechanisms that lead to the focusing in phononic crystal plates. One is based on the design of gradient index phononic crystal (GRIN PC) plate and the other one is based on the negative refraction effects. We numerically demonstrated the focusing behavior in the PC plates by the finite element method (FEM). For the case of the GRIN PC plate, the band structures and refraction indices (RI) of the GRIN PC plates with different filling fractions were calculated. And specific holes’ radii were designed to control the RI distribution to fit the hyperbolic secant curve. On the experimental side, focusing of the GRIN PC plates with two different scales was demonstrated. The experimental results show good agreements with those predicted from the numerical analyses. The second focusing mechanism is using the negative refraction effect in a PC plate which arranged in triangular lattice. In the analyses, the band structures and EFSs (equi-frequency surfaces) of the PC plates were calculated. The Snell’s law was introduced to predict the incident and refractive angles. Both the numerical analyses and experimental demonstrations were performed and the results also show good accordances.