Summary : The present doctoral thesis concerns the study of elastic and electromagnetic surface waves propagation in three-dimensional space (3-D) through anisotropic and unbounded media. It is composed of three distinctive parts under the headings: surface Green's dyads for a piezoelectric half-space; hybrid element method and application to SAW IDT modeling; and, electromagnetic surface waves on dielectric slabs.

Parts I and II come from an unique motivation, namely, the description of 3-D piezoelectric Surface Acoustic Waves (SAW) in Interdigital Transducers (IDTs). A typical SAW device consists of a piezoelectric anisotropic substrate on top of which sets of interleaved electrodes are deposed (IDT). Each IDT comprises a large number of electrodes with alternating voltages which, under certain geometric and material conditions, originate mechanical Rayleigh waves. As the wave propagates, a second set of IDT electrically picks up the filtered signal. To focus the acoustic beam in a certain direction, large electrode length-to-width ratios are employed. However, traditional numerical models for SAW IDTs have avoided dealing with such elongated structures by only considering the sagittal plane, using a FEM/BEM formulation. Today, stringent requirements on miniaturization and power consumption demand for smaller electrodes, and consequently, the neglected transversal effects come into play.

This research work sheds some light on these issues by proposing and succesfully implementing an ad hocsurface BEM formulation. Part I, concentrates on deriving the associated Green's functions while Part II shows the validity of a Galerkin mixed bases approach for this problem which can be extended to more general ones.

In Part III, we consider the unsolved questions of existence and uniqueness conditions for 3-D electromagnetic (EM) waves in isotropic layered media. Such structures are also known as dielectric slabs or open waveguides and are commonly found in optical integrated circuits. Although physically different, this problem constitutes an intermediary step to solve the same questions for the more complex cases of anisotropic media and elastic waves.Présentation Générale
General Outline
Part I: Surface Green's dyads for a Piezoelectric Half-Space
Notation
Introduction to SAW IDTs
1.1. What is a Surface Acoustic Wave Interdigital Transducer?
1.1.1. How do SAW IDTs work?
1.1.2. Applications of SAW components
1.1.3. SAW IDT Design: A very short glance
1.1.4. Why do we need to model SAW IDTs
1.2. The Physics behind SAW IDTs
1.2.1. Linear Elasticity
1.2.2. Waves in elastic media
1.2.3. Piezoelectricity
1.2.4. Waves in piezoelectric media
1.3. Surface acoustic waves
1.3.1. Choice of suitable modes
1.3.2. Classification of surface acoustic waves
1.3.3. Solution basis
1.3.4. Spectral surface Green's dyad
1.4. Modelling SAW IDTs: A Boundary Element Method perspective
1.4.1. Calculation of Green's functions
1.4.2. Boundary function bases
1.4.3. FEM/BEM models
1.5. Modelling 3-D SAW IDTs: The Challenge
2. Surface Green's Dyad for Non-Periodic Excitations
2.1. Piezoelectric half-space problem
2.1.1. Introduction
2.1.2. Model geometry
2.1.3. Master equations
2.1.4. Boundary conditions
2.1.5. Problem formulation
2.2. Spatial Surface Green's dyad for a single excitation
2.2.1. Introduction
2.2.2. Definition of the Green's dyad
2.2.3. Problem statement
2.2.4. Computation of the spectral surface Green's function
2.2.5. Isolation of singularities
2.2.6. Surface acoustic wave contribution G_SAW
2.2.7. Asymptotic behavior contribution G_infty
2.2.8. Contribution at the origin
2.2.9. Computation of G_reg
2.3. Future work
3. Surface Green's Dyad for x_1-periodic Excitations
3.1. Model geometry
3.2. Functional framework
3.2.1. Periodic distributions
3.2.2. Fourier series. Poisson's summation formulae
3.2.3. Periodic Sobolev spaces H^s_{1,p} and H^s_{1,p, loc}
3.3. Problem formulation
3.3.1. Piezoelectric x_1-periodic problem in R^3
3.3.2. 1-D periodic radiation conditions in R^3
3.3.3. Surface x_1-periodic Green's dyad
3.4. Computation of G^p
3.5. Calculation of G^p_SAW
3.5.1. Spectral representation in cartesian coordinates
3.5.2. Computation of line transforms for symmetric and antisymmetric parts
3.6. Calculation of G^p_{infty ,0}
3.6.1. Principal term G^{p,0}_{infty ,0}
3.6.2. Calculation of G}^{p,odd}_{infty ,0}
3.6.3. Calculation of G ^{p,even}_{infty ,0}
3.6.4. The limiting cases G_{infty ,0}(0,x_3;n)
3.7. Calculation of G^p_reg
3.8. Summary
Part II : Hybrid Elements Method: Modeling 3-D SAW IDTs
Notation
4. Logarithmic Integral Operators in Holder Spaces
4.1. Holder spaces
4.2. Cauchy-type integrals
4.3. Scalar Riemann-Hilbert problems
4.3.1. Homogeneous problem
4.3.2. Inhomogeneous problem
4.3.3. Hilbert problem
4.3.4. Cauchy integral problem
4.4. Application to logarithmic kernels
4.4.1. On a single segment
4.4.2. On many segments
4.4.3. Application to more general kernels
4.5. Summary
5. Logarithmic Integral Operators in Sobolev Spaces
5.1. Trigonometric bases and Sobolev spaces in [0,pi ]
5.1.1. Sobolev spaces in [0,pi]
5.1.2. Truncation error
5.2. Weighted L^2-spaces and Tchebychev polynomials
5.2.1. Link with H^s_j([0,pi]) spaces
5.3. Fourier-Tchebychev series
5.3.1. Truncation errors in the weighted L^2-norm
5.3.2. Truncation errors in the L^infty-norm
5.4. Application to logarithmic integral operators
5.4.1. On weighted L^2-spaces
5.4.2. On Sobolev spaces H^s_j([0,pi ])
5.4.3. Relation with the original operator L and Holder spaces
5.5. Summary
6. A Series of Electrostatic Problems
6.1. Isotropic problems in R^2
6.1.1. Model geometry
6.1.2. Functional spaces
6.1.3. Dirichlet problem
6.1.4. Neumann problem
6.1.5. Coercivity of the logarithmic integral operator and its inverse
6.1.6. An adapted spectral boundary element method
6.2. Isotropic problems in R^3
6.2.1. Flat edge problem
6.2.2}Bounded cylindrical screen
6.3. Interface problems
6.3.1. Isotropic materials in R^2
6.3.2. Isotropic materials in R^3
6.3.3. Anisotropic/isotropic materials in R^3
6.4. Final remarks
7. 3-D Electrostatic Hybrid Elements Model for SAW IDTs
7.1. Introduction
7.2. Charge distributions at Edges and corners
7.2.1. Edges
7.2.2. Corners
7.3. Hybrid Elements Model
7.4. Results and Discussion
7.4.1. Model validation
7.4.2. Mesh Design
7.4.3. Isotropic electrostatic Green's function
7.4.4. Anisotropic electrostatic Green's function
7.5. Conclusions
8. Full 3-D SAW IDT Model for Massless Electrodes
8.1. Introduction
8.2. Model Description
8.2.1. Asymptotic Green's dyad decomposition
8.2.2. Application of the Hybrid Element Method to G_44
8.3. Charge distribution in a single cell
8.4. Calculation of Mechanical Displacements
8.5. Conclusion and future work
Part III: Electromagnetic Surface Waves on Dielectric Slabs
Notation
9. Dielectric Slab Problem
9.1. Introduction
9.1.1. Existence results
9.1.2. Uniqueness results
9.1.3. Numerical issues
9.2. Problem formulation
9.2.1. Model geometry
9.2.2. Maxwell's equations
9.2.3. Functional spaces
9.2.4. Boundary conditions
9.2.5. Classical radiation conditions
9.2.6. Unperturbed scattering problem
9.3. Dyadic Green's function for isotropic materials
9.3.1. Transversal components
9.4. Polarization decomposition
9.4.1. TM polarization or E-modes
9.4.2. TE polarization or H-modes
9.5. General problem -- Spatial formulation
10. Spectral Green's Dyads
10.1. Surface Fourier transform
10.2. General problem - Spectral formulation
10.2.1. Solutions for the homogeneous equation in Omega _i
10.2.2. Square root determination
10.2.3. Solutions for the inhomogeneous equation in Omega _i
10.2.4. Reflection and transmission coefficients
10.2.5. Spectral solution for (P_g^{in})
10.2.6. Spectral solution for (P_g^{ext})
10.3. TM polarization - Spectral formulation
10.3.1. Spectral source and internal boundary conditions
10.3.2. Spectral Green's functions for (P_E^{in})
10.3.3. Spectral formulation for (P_E^{ext})
10.3.4. Transversal components
10.4. TE polarization - Spectral formulation
10.4.1. Spectral source and internal boundary conditions
10.4.2. Spectral formulation for (P_H^{in})
10.4.3. Spectral formulation for (P_H^{ext})
10.4.4. Transversal components
10.5. Summary191
10.5.1. Solution structure of interior excitation problems
10.5.2. Solution structure of exterior excitation problems
11. Existence of Surface Waves
11.1. Surface modes
11.1.1. Radiation modes
11.2. Discreteness of surface modes1
11.2.1. Symmetric slab
11.2.2. Asymmetric slab
12. Radiation Conditions -- Excitation Inside
12.1. Main results
12.2. Analysis outline
12.2.1. Calculation scheme
12.3. Green's function in Omega _1
12.3.1. Saddle point contribution
12.3.2. Surface mode or pole contribution
12.3.3. Branch point contributions
12.3.4. End point contributions
12.3.5. Summary
12.4. Green's functions in Omega _2
12.4.1. Surface mode contribution
12.4.2. Branch points contributions
12.4.3. End point contributions
12.4.4. Summary
12.5. Green's function in Omega _3
12.5.1. Stationary point contribution
12.5.2. Surface mode contribution
12.5.3. Branch points contributions
12.5.4. End point contributions
12.5.5. Summary
12.6. Dyad components asymptotics
12.6.1. Normal components G^{P,i}_{p,3j}
12.6.2. Transversal components G^{P,i}_{p,Tj} and G^{P,i}_{q,Tj}
12.7. Radiation condition proofs
12.7.1. Sommerfeld-type radiation conditions
12.7.2. Silver-Muller-type radiation conditions
12.8. Conclusion
Appendices
A. Physical Quantities
B. Dyads and tensors
C. A Few Concepts Used in SAW IDT Design
D. Fourier Transforms and Functional Spaces
E. Proofs for x_1-Periodic Asymptotic Contributions
F. Elements of Integral Operators
G. Derivation of Q^asym_alpha
H. Asymptotic Methods