Analyses of Bulk and Surface Acoustic Waves in 2D Phononic Crystal Waveguides Using FDTD Method

聲子晶體(phononic crystals)是一門結合了固態物理及彈性動力學(elastodynamic)特性的領域。聲子晶體結構由數種彈性材料在空間中週期性排列而成，聲波在此結構中傳遞時，由於受到的另一材料週期性出現影響而改變了原來的傳播現象。其中一個特別的性質便是頻溝(band gaps)的產生。頻溝指的是聲子晶體結構中的波傳模態出現不連續的現象，因此在該頻率範圍內聲波將無法傳遞。這樣的特性提供了利用聲子晶體發展控制聲波傳遞的迴路的基礎。本文中即針對聲波迴路中的重要元件—「波導(waveguides)」來進行分析，並探討在波導中控制波傳行為的技術。 本文中使用時間域有限差分法(finite-difference time-domain, FDTD)來進行研究。時間域有限差分法可對多樣化的問題進行計算，因此廣泛地被使用在波傳的研究中。在本文中推導了差分方程式，並對必須使用的邊界條件加以介紹，包含了布拉克(Bloch)週期條件及完全匹配層(perfectly matched layer, PML)條件。應用這些條件，時間域有限差分法不僅可以計算暫態的波傳行為，可更進一步分析聲子晶體週期性結構的頻散(dispersion)關係。然而由於在進行表面波波傳計算時將會耗費大量的計算，因此研究中建立了一套叢集式個人電腦系統(PC Cluster system)，並配合訊息傳遞介面(message passing interface, MPI)函式庫撰寫平行化程式並進行計算。透過平行計算的協助，表面波在波導內傳遞的計算速度提昇，才使得此項研究變為可能。 研究中首先呈現利用時間域有限差分法計算頻散關係的方法，然後對鋼/環氧樹脂、鎢/矽及砷化鋁/砷化鎵組成之聲子晶體進行計算，呈現這些聲子晶體內徹體波及表面波的模態。其中在鋼/環氧樹脂及鎢/矽聲子晶體中皆存在有聲波的完全頻溝(complete band gap)，而聲子晶體波導便是利用完全頻溝設計出來的。聲子晶體波導可由結構連續的點缺陷相連而成，而要分析這樣的結構，則須利用超晶格(supercell)的方法。透過超晶格，聲子晶體波導內的徹體波及表面波模態都被計算出來，同時也可觀察到波導內波傳的現象。更進一步的，波導內耦合(coupling)效應與波導寬度的影響也可加以計算。同時利用上述分析的結果，設計了聲波耦合器(coupler)，可在聲波通道中分離出特定頻率的波。此外，針對表面波在急轉彎的波導中傳遞，也提出了一個改良的設計以提升其效率。綜言之，本研究中針對聲子晶體表面波的研究呈現一套完整的分析設計程序，可作為未來發展聲波迴路的基礎。Phononic crystal research covers both of the fields of solid-state physics and elastodynamic theory. A phononic crystal is constructed of multi-types elastic materials arranged periodically in space, and acoustic waves propagating in it is affected by the periodic scatters. A specific property is band gaps resulted from the discontinuity of eigenmodes, and thus propagation of acoustic waves is forbidden in this frequency range. This property inspires the attempts to control acoustic waves and develop the acoustic circuit engineering. In this text, one of the important components – phononic crystal waveguide is analyzed and the techniques of manipulating waves in the waveguides are discussed. In this thesis, the finite-difference time-domain (FDTD) method is adopted. FDTD method is a flexible tool and widely used in the study of wave propagations. In this study, the difference equations are derived and the necessary boundary conditions, i.e., Bloch’s periodic condition and perfectly matched layer (PML), are introduced. With these conditions, the FDTD can not only calculate the transient wave propagation but also analyze the dispersion of the periodic structures. Besides, analyzing surface waves consumes huge computation power, and therefore a PC Cluster system is established to serve parallel calculation. A parallel FDTD program is realized with message passing interface (MPI) library. With the PC cluster and parallel calculation, study of surface waves inside waveguides is accelerated and becomes feasible. The technique of analyzing dispersions of phononic crystals using FDTD method is presented firstly. The properties of steel/epoxy, tungsten/silicon and AlAs/GaAs phononic crystals are calculated. The eigenmodes of both BAW and SAW are calculated and complete band gaps are obtained in the steel/epoxy and tungsten/silicon cases. Based on the complete band gaps, phononic crystal waveguides are designed by connecting point defects in phononic crystals. To investigate the defect modes inside the waveguides, the supercell technique is used and the dispersion relations of bulk and surface waves are obtained. The guided waves inside waveguides are investigated and the propagation is demonstrated. Further, the effect of coupling waveguides and the waveguide width is discussed. Then a waveguide coupler is designed to select the wave of a specific wavelength in a dual waveguides system. Besides, an improved bending waveguide is reported to improve the transmission of surface waves. To sum up, a procedure of analyzing phononic crystals and designing waveguides has been presented in this work and the results can serve as an important foundation for the future development of acoustic circuits.謝誌 I ABSTRACT II 中文摘要 III CONTENTS IV LIST OF FIGURES VI LIST OF TABLES X NOTATIONS XI CHAPTER 1 INTRODUCTION 1 1.1 Motivation 1 1.2 Literature Review 2 1.3 An Overview of the Text 6 CHAPTER 2 FINITE-DIFFERENCE TIME-DOMAIN METHOD 10 2.1 Elastodynamic Equation 10 2.2 Three-Dimensional Difference Equation 12 2.2.1 Taylor Expansion 12 2.2.2 Finite-Difference Expression for Elastodynamic Equations 15 2.3 Numerical Dispersion and Stability 18 2.3.1 Numerical Wave Vector and Frequency Expressions 19 2.3.2 Numerical Dispersion 21 2.3.3 Stability Condition 23 2.4 Boundary Conditions 27 2.4.1 Free Boundary Condition 27 2.4.2 Periodic Condition 29 2.4.3 Perfectly Matched Layer Condition 32 CHAPTER 3 PARALLEL COMPUTING IN THE PC CLUSTER SYSTEM 40 3.1 PC Cluster Systems 41 3.1.1 The Basic of Computers 41 3.1.2 The Architecture of High Performance Computers 42 3.1.3 The Architecture of PC Cluster Systems 45 3.2 Massage Passing Interface Library 47 3.3 Parallel FDTD Program 50 3.3.1 Parallelism Model 50 3.3.2 Development of Parallel FDTD Program 52 CHAPTER 4 ANALYSIS OF EIGEN MODES OF PHONONIC CRYSTALS USING FDTD 70 4.1 Property of Periodic Structure 70 4.2 Bulk Acoustic Waves inside Two-Dimensional Phononic Crystal 73 4.2.1 Steel/Epoxy Square Lattice Phononic Crystal 74 4.2.2 Tungsten/Silicon Square Lattice Phononic Crystal 76 4.3 Surface Acoustic Waves inside Phononic Crystals 78 4.3.1 AlAs/GaAs Square Lattice Phononic Crystal 79 4.3.2 Steel/Epoxy Square Lattice Phononic Crystal 81 4.3.3 Tungsten/Silicon Square Lattice Phononic Crystal 82 4.4 Three-Dimensional Phononic Crystals 83 4.4.1 Simple-Cubic Lattice Phononic Crystals 84 4.4.2 Body-Centered Cubic Lattice Phononic Crystals 84 4.4.3 Face-Centered Cubic Lattice Phononic Crystals 84 CHAPTER 5 PHONONIC CRYSTAL WAVEGUIDES 107 5.1 Defect Modes and Supercell Method 107 5.2 Bulk Wave Waveguides 110 5.2.1 Defect Modes of BAW in Phononic Crystal Waveguides 110 5.2.2 Coupling Effect and Supermodes of Coupling Phononic Waveguides 112 5.3 Surface Wave Waveguides 114 5.4 Modulation of Phononic Crystal Waveguides 116 5.4.1 The Width of Waveguides 116 5.4.2 Analysis of an Elastic Wave Coupler 117 5.4.3 An Improved Surface Waveguide with a Sharp Bend 120 CHAPTER 6 CONCLUSIONS AND OUTLOOK 143 Appendix A. Finite-Difference approximations of PML condition 146 REFERENCE 15