Phononic crystals are artificial periodic structures that can alter efficiently the flow of sound, acoustic waves, or elastic waves. They were introduced about twenty years ago and have gained increasing interest since then, both because of their amazing physical properties and because of their potential applications. The topic of phononic crystals stands as the cross-road of physics (condensed matter physics, wave propagation in inhomogeneous and periodic media) and engineering (acoustics, ultrasonics, mechanical engineering, electrical engineering). Phononic crystals cover a wide range of scales, from meter-size periodic structures for sound in air to nanometer-size structures for information processing or thermal phonon control in integrated circuits. Phononic crystals have a definite relation with the topic of photonic crystals in optics. The marriage of phononic and photonic crystals also provides a promising structural basis for enhanced sound and light interaction.
As the topic is getting popular, it is nowadays presented and discussed at various international conferences. After the first ten years during which the topic has remained mainly theoretical with a few proof-of-concept demonstrations in the literature, the evolution has been towards applications, instrumentation, and novel designs. The physical explanations for various effects are now well understood and efficient numerical methods and analysis tools have been developed.
The book contains a comprehensive set of finite element model (FEM) scripts for solving basic phononic crystal problems. The scripts are short, easy to read, and efficient, allowing the reader to generate for him(her)self band structures for 2D and 3D phononic crystals, to compute Bloch waves, waveguide and cavity modes, and more.
Vincent Laude, Centre National de la Recherche Scientifique, Besançon cedex, France.
1 Introduction [6 p.]
Description and purpose of the book. Introduction of some elementary concepts. History of the phononic crystal
2 Waves in periodic media [40 p.]
A presentation of waves in periodic media devoid of complications like polarization, anisotropy, tensors, loss, etc.
Self-contained presentation for scalar waves.
2.1 Bloch theorem
Scalar wave theory. Scalar Helmholtz equation. Bloch theorem.
2.2 Physical origin of band gaps
1D periodic media. Scattering and diffraction. Bragg band gaps. Local and Fano resonances.
2.3 Brillouin zone
Definition. Direct and reciprocal lattice.
2.4 The band structure
Fourier transforms. Wave vectors. Band structure. Dispersion, group velocity. Equifrequency contours. Analogy
with phonons in atomic lattices.
2.5 Appendix: Brillouin zones for 2D and 3D lattices
Geometrical description of the most common lattices.
3 Acoustic waves [20 p.]
A synthetic presentation of the subject, with reference to other basic books.
3.1 Dynamic equations
Particle velocity and pressure. Acoustic equations.
3.2 Constants of fluids
Constants for fluids. Determination of bulk velocities.
3.3 Loss and viscosity
Representation of propagation loss in fluids. Modifications of equations (complex material constants).
3.4 Reflection and refraction
Brief review of reflection and refraction at the interface of 2 media. Fresnel formulas.
4 Sonic crystals [50 p.]
Introduce sonic crystals (that can be described by pressure waves), with accent on finite element modeling and
basic properties.4.1 Modeling of sonic crystals
4.1.1 Analysis via plane wave expansion (PWE)
4.1.2 Multiple scattering theory (MST and LMS)
4.1.3 Finite element modeling (FEM)
4.2 2D sonic crystal
Steel cylinders in air. Steel cylinders in water. Measurement techniques. Comparison with experiment. Deaf
4.3 3D sonic crystals
Steel beads in water.
4.4 Tutorial: sonic crystal analysis with FEM
Generation of band structures. Plotting Bloch waves. Worked examples with ff++.
4.5 Appendix: Weak form modeling of sonic crystals. Lagrange Finite elements. Bloch
waves and FEM.
5 Elastic waves [40 p.]
A synthetic presentation of the subject, with reference to other basic books, plus an original part on FEM
5.1 Dynamic equations
Strain and Stress. Elastic constants. Elastodynamic equations.
5.2 Christoffel equation for bulk waves
Anisotropy of wave propagation in crystalline solids. Slowness curve. Wave surface. Polarization. Group velocity.
Poynting theorem and energy conservation.
5.3 Piezoelectric media
Description of the effect. Generalization of the concepts of the previous section.
5.4 Plate waves
Lamb and other plate waves. Dispersion diagram.
5.5 Surface waves
Rayleigh and other surface waves. Radiation and leakage. Slowness curves for SAW.
5.6 Tutorial: modeling plate waves with FEM
5.7 Appendix: tensors
6 Phononic crystals for bulk waves [50 p.]
6.1 Modeling of phononic crystals
6.1.1 Analysis via plane wave expansion (PWE)
6.1.2 Finite element modeling (FEM)
6.2 2D phononic crystal
Holey and solid-solid PC, for most common material combinations. Comparison with experiments.
6.3 3D phononic crystals
Steel beads in epoxy. Comparison with experiments.
6.4 Tutorial: phononic crystal analysis with FEM
Generation of band structures. Plotting Bloch waves. Worked examples with ff++.6.5 Appendix: weak form modeling of phononic crystals
7 Phononic crystals for surface and plate waves [40 p.]
Presentation based on PWE and/or FEM. Surface boundary conditions and determinants.
7.2 Phononic plates
Specific properties and discussion of various forms. Preferred example: holey silicon plate.
7.3 Surface phononic crystals
Specific properties and discussion of various forms. Preferred examples: holey silicon and lithium niobate. The
sound cone and leakage.
7.4 Measurement methods
Electrical transduction. Optical transduction. Optical measurement of surface displacements. Comparison with
7.5 Tutorial: phononic plates and surface waves with FEM
8 Coupling of acoustic and elastic waves in phononic crystals [20 p.]
8.1 Phononic crystal of solid inclusions in fluid
8.2 Phononic plates in water and air
8.3 Corrugated surfaces and plates