Analysis of Electromagnetic and Elastic Wave Propagation in Multilayered Anisotropic Structures

This work essentially deals with a numerical approach, the Spectral Element Method, that constitutes a valid and effective alternative to the classical transverse resonance method, generally employed to study layered guiding structures.

In the first chapter the state of art is resumed, referring to some of the many published works appearing in literature, so outlining the main features of the transverse analysis and the corresponding drawbacks. The practical applications of layered structures are recalled and then the usefulness of knowing the modal spectra of such waveguides is shown.

Successively, in view of studying both electromagnetic and acoustic wave propagation, the problem is formulated in an abstract form as a general linear differential equation comprising an eigenvalue. After observing that layered structures are mathematically described by parameters that are piecewise constant functions of the space variables, the need for expansion functions defined on subdomains, which is the key feature of the SEM, is explained. Hence the SEM is described in details, pointing out its advantages and showing by means of a simple problem why an exponential convergence must be expected.

In Chapter 2 the basic hypotheses and equations of the acoustic field theory are presented and then they are used as a starting point to derive the proper wave equations in the case of isotropic and anisotropic layered waveguides. Then, the application of the SEM is developed step by step.

In Chapter 3 first a layered waveguide comprising only isotropic media is studied, then the Maxwell's equations are rearranged in order to derive the suitable formulation of the problem for general anisotropic structures, transversally limited with various boundary conditions. The mode orthogonality is analytically proven when the anisotropy is only dielectric (or magnetic) and confined in a plane normal to the waveguide axis.

Successively, a first extension of the standard SEM is introduced. In Chapter 4 the problem of finding guided modes in open planar structures is tackled and solved using suitable expansion functions in the unbounded regions in order to model the damped fields.

Finally, in Chapter 5 the standard SEM is applied to find modes in cylindrical layered waveguides comprising isotropic media, after writing two wave equations for the longitudinal fields.

Every chapter ends presenting a study of the convergence properties of the SEM; in particular the exponential rate of convergence is justified in all practical cases and normalized error plots, which are useful in evaluating and predicting the error committed on the approximated solutions, are obtained.

Numerical results such as dispersion curves and field profiles are also reported as some example of application.

Two Appendices complete the work, providing the main formulae relative to Legendre and Laguerre polynomials and the matrices that allow to implement the SEM automatically.