Phononic Crystals.pdf

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Preface

Contents

1 Introduction to Acoustics of Phononic Crystals. Homogenization at Low Frequencies

1.1 Introduction

1.2 Homogenization: Quasi-Static Limit and the Method of Plane Waves

1.2.1 One-Dimensional Periodicity

1.2.2 Two-Dimensional Periodicity

1.2.3 Three-Dimensional Periodicity

1.3 Homogenization: The Multiple-Scattering Method

1.3.1 Homogenization of Mixed Lattices

1.3.2 Homogenization of a Medium with Local Resonances

References

2 Fundamental Properties of Phononic Crystal

2.1 Introduction to the Concept of Phononic Crystals and Their Band Structures

2.2 Dispersion Curves and Band Gaps in 2D Phononic Crystals

2.2.1 Origin of the Band Gaps: Bragg Gaps and Local Resonances

2.2.2 Behavior of the Band Gaps as a Function of the Geometrical and Physical Parameters

2.2.3 Solid–Solid Phononic Crystal

2.2.4 Solid–Fluid Phononic Crystal

2.2.5 Fluid–Fluid Phononic Crystal

2.2.6 Locally Resonant Phononic Crystal

2.3 Localized Modes Associated with Defects

2.3.1 Guiding

2.3.2 Filtering

2.3.3 Demultiplexing

2.3.4 Tunability

2.4 Concluding Remarks and Further Developments in the Field of Phononic Crystals

References

3 The Three-Dimensional Phononic Crystals

3.1 Introduction

3.2 Phononic Lattices

3.2.1 Multiple Scattering and the LMS Method

3.2.2 Formation of Omnidirectional Frequency Gaps, Attenuation, and Tunneling

3.2.3 Next-Generation 3D Phononic Structures

3.3 Imperfect Phononic Structures: From Periodicity to Disorder

3.3.1 Layered Non-periodic Heterostructures

3.3.1.1 Planar Defects

3.3.1.2 Gradient Layered Structures

3.3.1.3 Planar Disorder

3.3.2 Linear Defects

3.3.2.1 Multiple Scattering in the Real Space

3.3.2.2 Waveguiding

3.4 Multi-component 3D Phononic Crystals (Locally Resonant and Acoustic Metamaterials)

3.4.1 Locally Resonant Phononic Crystals

3.4.1.1 Bulk Structures

3.4.1.2 Finite Plates

3.4.2 Acoustic Metamaterials

References

4 Computational Problems and Numerical Techniquesfor the Analysis of Phononic Crystals

4.1 Basic Equations for Wave Propagation

4.1.1 Equations for Solids

4.1.2 Equations for Fluids

4.1.3 Consideration of Material Loss

4.2 Computational Problems of Phononic Crystals

4.2.1 Classification by Geometry

4.2.2 Classification by Problem Type

4.3 Multiple Scattering Theory and Layer Multiple Scattering Methods

4.4 Plane Wave Expansion Method

4.4.1 Band Structures with PWE

4.4.2 Evanescent Waves in Phononic Crystals

4.5 Finite Element Method

4.6 Finite-Difference Time-Domain Method

4.6.1 Boundary Conditions

4.6.2 Calculation of Dispersion Relations

4.6.3 Calculation of Transmission Spectra

4.7 Conclusion

References

5 Phononic Crystal Membranes (Slabs or Plates)

5.1 Introduction

5.1.1 A Brief History of the PnC Slab Structures

5.1.2 Problem of Leaky SAWs in PnC Structures

5.2 PnC Slab (Membrane) Structures

5.2.1 Types of PnC Slab Structures Based on Geometry

5.2.2 Inclusion-Based PnC Slabs

5.2.3 Decorated Stubs PnC Membranes

5.2.4 Examples of PnC Slab Structures

5.3 Methods for the Analysis of PnC Slabs

5.3.1 FDTD for the Analysis of PnC Slabs

5.3.1.1 Free Surface Boundary Condition

5.3.2 PWE for the Analysis of PnC Slabs

5.3.3 FE Method for the Analysis of PnC Slabs

5.3.4 Other Significant Techniques for the Analysis of PnC Slab Structures

5.3.4.1 Transfer, Reflection, and Transmission Matrix Methods

5.3.4.2 Scattering Matrix (Modal) Method

5.4 Fabrication and Characterization of PnC Slab Structures

5.5 Case Study: Engineering in PnC Slabs with Void Inclusions

5.5.1 Design of Void/Solid Si PnC Slab Structures with High-Frequency Complete PnBGs

5.5.2 Fabrication and Characterization of Si PnC Slab Structures

5.6 PnC Crystal Slab Devices and Band Gap Engineering

5.6.1 Energy Confinement in MM Resonators in PnC Slabs

5.6.1.1 MM PnC Slab Resonators with Resonant Tunneling Excitation

5.6.2 Support Loss Suppression of MM Resonators Using PnBG Structures

5.6.2.1 Support Loss in MM Resonators

5.6.2.2 A Support Loss Suppressed PnC Slab Resonator

5.6.3 PnC Slab Waveguides

5.6.3.1 An Efficient PnC Slab Waveguide

5.6.4 High-Q Waveguide-Based PnC Resonators

5.7 Further Research Trends and Perspectives of PnC Slab Structures

5.7.1 Dispersive PnC Slab Structures

5.7.2 Optomechanical Crystal Slabs

5.7.3 Thermal Phonons Control in PnC Slabs

References

6 Surface Acoustic Waves in Phononic Crystals

6.1 Introduction

6.2 Theoretical Formulations

6.2.1 PWE Method

6.2.1.1 General Equations

6.2.1.2 Case of Void Inclusions in a Solid Matrix

6.2.2 Finite-Difference Time-Domain Method

6.3 SAWs in Phononic Crystals

6.3.1 Frequency Band Structures

6.3.2 Band Gaps and SAW Characteristics

6.3.3 Bleustein–Gulyaev Waves

6.4 Phononic Crystal Waveguides for SAW

6.4.1 Complete Band Gaps of Steel/Epoxy Phononic Crystals

6.4.2 SAWs Inside a Phononic Crystal Waveguide

6.5 Experiment on the SAW Band Gap

6.5.1 Band Gap for SAWs in Silicon-Based Phononic Crystals

6.5.2 Piezoelectric Phononic Crystals

6.6 Application of SAW Band Gap to SAW Devices

6.6.1 Reflective Grating for SAWs Using Phononic Crystals

6.6.2 SAW Resonator

6.7 Conclusions

References

7 Optical Characterization of Phononic Crystalsin the Time Domain

7.1 Introduction

7.2 Experimental Setup for Scanning Laser Interferometry in the Time Domain

7.2.1 Optical Pump-Probe Technique

7.2.2 Interferometer

7.2.3 Scanning System

7.3 Applications to Phononic Crystals

7.3.1 From Time-Resolved Data to the Acoustic Dispersion Relation

7.3.2 One-Dimensional Phononic Crystals

7.3.3 Two-Dimensional Phononic Crystals

7.3.4 Summary: Capabilities and Limitations

7.4 Conclusions

References

8 Optical Characterization of Phononic Crystalsin the Frequency Domain

8.1 Introduction

8.2 Scanning Laser Interferometry in Frequency Domain

8.2.1 Homodyne Detection

8.2.2 Heterodyne Detection

8.3 Capabilities and Limitations of Scanning Laser Interferometers

8.4 Applications to Phononic Crystals

8.4.1 Surface Acoustic Waves in Micro-Structured Acoustic Metamaterials

8.4.1.1 Extracting Propagation Information from Random SAW Field

8.4.1.2 Two-Dimensional Phononic Crystal for 200MHz SAWs

8.4.2 Bulk Acoustic Waves in Acoustic Metamaterials

8.4.2.1 Thin-Film Mirror Transfer Properties

8.4.2.2 Further Use of Dispersion Diagrams in Data Analysis

8.5 Conclusion

References

9 Future Prospects of Phononic Crystals and PhononicMetamaterials

9.1 Introduction

9.1.1 Timing Elements Using Phononic Crystals

9.1.2 Signal Processing Functions for Communications

9.1.3 Phononic Crystal Sensors

9.1.4 Negative Refraction and Superlensing

9.1.5 Phononic Crystals for Opto-Mechanics

9.1.6 Phononic Crystals for Liquid Control and Handling

9.1.7 Nonlinearity Effects in Phononic Crystals

9.1.8 Energy Scavenging Using Phononic Crystals

9.1.9 Thermal Phonon Control Through the Use of Phononic Crystals

9.1.10 Noise Control and Sound Proofing

9.1.11 Phononic (Acoustic or Thermal) Diodes and Transistors

9.1.12 Immunization to Environmental Variables

9.1.13 Note on Acoustic Metamaterials

9.2 Conclusion

References

AbdelkrimKhelif· AliAdibi Editors Phononic Crystals Fundamentals and Applications

Phononic Crystals

Abdelkrim Khelif • Ali Adibi Editors Phononic Crystals Fundamentals and Applications 123

Editors Abdelkrim Khelif Institut FEMTO-ST Centre National de la Recherche Scientiﬁque Besançon Cedex, France Ali Adibi School of Electrical Engineering Georgia Institute of Technology Atlanta, GA, USA ISBN 978-1-4614-9392-1 DOI 10.1007/978-1-4614-9393-8 ISBN 978-1-4614-9393-8 (eBook) Library of Congress Control Number: 2015943375 Springer New York Heidelberg Dordrecht London © Springer Science+Business Media New York 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Science+Business Media LLC New York is part of Springer Science+Business Media (www. springer.com)

Preface Phononic crystals (PnCs) are novel synthetic periodic materials for controlling and manipulating the propagation of elastic (or acoustic) waves. The periodic nature of PnCs gives them novel properties that cannot be found in bulk materials. For example, PnCs can exhibit acoustic (or phononic) bandgaps, which are frequency ranges in which the propagation of acoustic waves inside the PnC is prohibited. The addition of defects to a perfect PnC with a phononic bandgap allows for the design of devices like waveguides and cavities to control the propagation of acoustic waves inside the bandgap and to enable novel functionalities in a very compact structure. Imminent impact of PnCs is expected in the near future in applications like wireless communications, sensing, acoustic signal processing, and ultrasound imag- ing. Novel devices (such as acoustic ﬁlters, resonators, sources and lenses) with outstanding performance measures are being enabled by the use of PnCs. In addition, the use of these structures to form acoustic metamaterials can uncover novel effects like negative refraction, acoustic invisibility, or superlensing. This, in turn, can enable researchers to design functional structures with such performance that cannot be obtained with conventional acoustic materials. While the research in the ﬁeld of phononic crystals and acoustic metamaterials is at the early stages, their optical counterparts (i.e., photonic crystals) have already been demonstrated to possess unique properties that are not achieved using conventional bulk materials. The properties of photonic crystals have been the subject of intensive investigations in the last decade, and several successful books have been published to address their unique properties and applications. Knowing that the research in PnCs is in its infancy, and more attention is given to this ﬁeld lately, the ﬁeld is expected to expand considerably in the next few years. The purpose of this book is to present a detailed overview of the state of the ﬁeld from material, device, and application perspectives, and provide the necessary tools for researchers to explore the ﬁeld. To achieve this goal, this book covers the simulation, fabrication, and characterization methods used to design and experiment with PnCs to the level that is accessible for both the experienced and beginner in the ﬁeld. The book also reports the most important advances in the ﬁeld in the last few years. v

vi Preface The idea for this book ﬁrst came up in summer 2009, where we co-chaired the ﬁrst International Workshop on Photonic Crystals (Nice, France, 2009), in which all experts in the ﬁeld were invited. The need for an all-encompassing reference in the ﬁeld of phononic crystals was recognized in the meeting. After that meeting, we spent an extensive amount of time looking into the needs of the community to form the structure of the chapters in the book and to convince the experts in the ﬁeld (who were among the participants in the workshop) to write their respective chapters. The authors of these chapters are among the world leaders in their respective ﬁelds with years of experience in performing cutting-edge research and educating young scientists and engineers. In addition to presenting the landscape of the research in this ﬁeld, we hope that this book can provide interested readers with an in-depth knowledge of the ﬁeld. The individual chapters are written in such a way that they can be used as the text material for enhancing graduate-level courses in mechanical or electrical engineering disciplines. At the end of this journey, we would like to thank all those who helped us in forming this book through their discussions, contributions to the book, and reviews of the different sections. We also like to thank the many researchers (students, postdocs, members of technical staff, and professors) whose contributions are covered in this book. Our special thanks go to Dr. Ali A. Eftekhar for his key role in forming the idea of the book, his help in deﬁning different chapters, and his excellent feedback at different stages of forming the book. Besançon Cedex, France Atlanta, GA, USA March 2015 A. Khelif A. Adibi

Contents 1 Introduction to Acoustics of Phononic Crystals. Homogenization at Low Frequencies ...................................... José Sánchez-Dehesa and Arkadii Krokhin 1 2 Fundamental Properties of Phononic Crystal ............................ 23 Yan Pennec and Bahram Djafari-Rouhani 3 The Three-Dimensional Phononic Crystals ............................... 51 Badreddine Assouar, Rebecca Sainidou, and Ioannis Psarobas 4 Computational Problems and Numerical Techniques for the Analysis of Phononic Crystals ..................................... Vincent Laude and Abdelkrim Khelif 85 5 Phononic Crystal Membranes (Slabs or Plates) .......................... 109 Saeed Mohammadi and Ali Adibi 6 Surface Acoustic Waves in Phononic Crystals ............................ 145 Tsung-Tsong Wu, Jin-Chen Hsu, Jia-Hong Sun, and Sarah Benchabane 7 Optical Characterization of Phononic Crystals in the Time Domain .......................................................... 191 Osamu Matsuda and Oliver B. Wright 8 Optical Characterization of Phononic Crystals in the Frequency Domain ................................................... 215 Kimmo Kokkonen 9 Future Prospects of Phononic Crystals and Phononic Metamaterials ................................................................ 239 Saeed Mohammadi, Abdelkrim Khelif, and Ali Adibi vii

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