Paul Gabrielli

Surface polaritons on left-handed cylinders: A complex angular momentum analysis

We consider the scattering of electromagnetic waves by a left-handed cylinder -- i.e., by a cylinder fabricated from a left-handed material -- in the framework of complex angular momentum techniques. We discuss both the TE and TM theories. We emphasize more particularly the resonant aspects of the problem linked to the existence of surface polaritons. We prove that the long-lived resonant modes can be classified into distinct families, each family being generated by one surface polariton propagating close to the cylinder surface and we physically describe all the surface polaritons by providing, for each one, its dispersion relation and its damping. This can be realized by noting that each surface polariton corresponds to a particular Regge pole of the

Whispering-gallery modes and resonances of an elliptic cavity

S

Exponentially improved asymptotic expansions for resonances of an elliptic cylinder

matrix of the cylinder. Moreover, for both polarizations, we find that there exists a particular surface polariton which corresponds, in the large-radius limit, to the surface polariton which is supported by the plane interface. There exists also an infinite family of surface polaritons of whispering-gallery type which have no analogs in the plane interface case and which are specific to left-handed materials.

Exact S-matrix for N-disc systems and various boundary conditions: I. Generalization of the Korringa-Kohn-Rostoker-Berry method

Eigenfunctions of the whispering-gallery type in elliptic cavities are considered. Asymptotic expansions for resonances are derived from the uniform asymptotic expansions of Mathieu functions and modified Mathieu functions constructed by applying the Langer-Olver method. These asymptotic expansions are improved by including exponentially small terms which lie beyond all orders of the perturbative series and can be captured by carefully taking into account Stokess phenomenon. A classification of resonances along the four irreducible representations of 2v (the symmetry group of the elliptic cavity) is provided, and the splitting up of resonances is then understood in connection with the breaking of O(2)-symmetry (invariance under any rotation).

Acoustic scattering by elastic cylinders of elliptical cross-section and splitting up of resonances

Scattering by an elliptic cylinder is considered. Asymptotic expansions for Regge poles and resonances are derived from the uniform asymptotic expansions of Mathieu functions and modified Mathieu functions constructed by applying the Langer-Olver method. In addition, asymptotic expansions for resonances are exponentially improved by emphasizing the role of the symmetries of the scatterer. The splitting up of resonances is then explained in terms of the symmetry breaking O(2)2v.

EXACT S-MATRIX FOR N-DISC SYSTEMS AND VARIOUS BOUNDARY CONDITIONS: II. DETERMINATION AND PARTIAL CLASSIFICATION OF RESONANCES

Scattering of waves and particles from N discs fixed in a plane is studied. Various boundary conditions on the scatterers, corresponding to mesoscopic quantum physics, acoustics or electromagnetism, are considered. An exact formalism allowing us to calculate the S-matrix, its scattering resonances and far-field form functions is developed for systems with discrete , and symmetries. It extends the Korringa-Kohn-Rostoker method developed in the context of solid state physics and generalizes the works of Berry and of Gaspard and Rice in quantum chaos.

Elastodynamics and resonances in elliptical geometry

The scattering of a plane acoustic wave by an infinite elastic cylinder of elliptical cross section is studied from a modal formalism by emphasizing the role of the symmetries. More precisely, as the symmetry is broken in the transition from the infinite circular cylinder to the elliptical one, the splitting up of resonances is observed both theoretically and experimentally. This phenomenon can be interpreted using group theory. The main difficulty stands in the application of this theory within the framework of the vectorial formalism in elastodynamics. This method significantly simplifies the numerical treatment of the problem, provides a full classification of the resonances, and gives a physical interpretation of the splitting up in terms of symmetry breaking. An experimental part based on ultrasonic spectroscopy complements the theoretical study. A series of tank experiments is carried out in the case of aluminium elliptical cylinders immersed in water, in the frequency range 0kr50, where kr is the reduced wave number in the fluid. The symmetry is broken by selecting various cylinders of increasing eccentricity. More precisely, the greater the eccentricity, the higher the splitting up of resonances is accentuated. The experimental results provide a very good agreement with the theoretical ones, the splitting up is observed on experimental form functions, and the split resonant modes are identified on angular diagrams.

Diffraction of ultrasonic waves from elastic Galois gratings

Scattering of waves and particles from two- and three-disc systems with discrete and symmetries is studied for various boundary conditions. Resonances are numerically determined, and partially classified by using the irreducible representations of the symmetry groups. New physical effects are expected (splitting up of resonances and resonances of interaction between the scatterers). Some of these effects can be observed on the far-field form functions.

Scattering by immersed two‐dimensional elastic galois gratings

The resonant modes of two-dimensional elastic elliptical objects are studied from a modal formalism by emphasizing the role of the symmetries of the objects. More precisely, as the symmetry is broken in the transition from the circular disc to the elliptical one, the splitting up of resonances and level crossings are observed. From the mathematical point of view, this observation can be explained by the broken invariance of the continuous symmetry group associated with the circular disc. The elliptical disc is however invariant under the finite group and the resonances are classified and associated with a given irreducible representation of this group. The main difficulty arises in the application of the group theory in elastodynamics where the vectorial formalism is used to express the physical quantities (elastic displacement and stress) involved in the boundary conditions. However, this method significantly simplifies the numerical treatment of the problem which is uncoupled over the four irreducible representations of . This provides a full classification of the resonances. They are tagged and tracked as the eccentricity of the elliptical disc increases. Then, the splitting up of resonances, which occurs in the transition from the circular disc to the elliptic one, is emphasized. The computation of displacement normal modes also highlights the mode splittings. A physical interpretation of resonances in terms of geometrical paths is provided.

Splitting up resonances of elastic elliptical disc

A significant attenuation of the specular echo in a wide frequency domain occurs in scattering of waves by Galois gratings.Diffraction of ultrasonicwaves by immersed one‐dimensional elasticgratings constructed from the finite Galois fields GF(p m ) is investigated. We develop a theoretical model taking into account the elasticity of the structure, its finite thickness, the boundary conditions at the fluid–solid and solid–fluid interfaces, and the periodicity of the gratings. In addition to the typical attenuation of the specular echo predicted by discrete Fourier transform, such a model predicts minima in the spectrum of the reflection coefficient corresponding to the excitation of surface waves (generalized Rayleigh and Scholte–Stoneley waves) by mode conversion and to the excitation of Lamb‐type modes. The attenuation as well as the minima are experimentally observed and these results are in good agreement with theoretically predicted ones. Galois gratings could constitute an advance in the search for new anechoic devices.