The concept of Feshbach resonances, developed for quantum mechanical scattering, is applied in the analysis of elastic wave scattering oﬀ Phononic crystal slabs. For the purpose of illustration and as an example, we will consider the so-called grating waveguide structure (GWS) which consists of a plane waveguide with a one-dimensional Phononic crystal slab above it. An elastic wave is incident at the Phononic slab and generates guided modes inside the waveguide. The wave also produces reflected and transmitted waves at the boundaries of the slab. The fundamental mechanism for the formation of resonances is the process of turning a bound state (guided mode) into a resonance state. A Feshbach resonance occurs if, by neglecting certain couplings, a bound state of the entire system exists. Accounting for the couplings converts this state into a resonance. In the Feshbach Projection Operator (FPO) formalism, the full function space is divided into two subspaces: the Q subspace contains all wave functions that are localized inside the idealized closed system and vanish outside of it while the wave functions of the P subspace are extended up to infinity. P and Q are interpreted as the projectors on the background and the resonant subspaces respectively. The projectors P and Q are not defined within the Feshbach method. This makes the method very general and, on the other hand, it introduces some arbitrariness which is difficult to control. Some authors define first the P space as an approximation to the background scattering but most authors define first the Q space as an approximation to the resonance. This step relies always on some physical intuition which is not easy to quantify. In practice, P or Q should be chosen such that the resulting background scattering phase or cross-section is slowly depending on the scattering energy in the neighborhood of the resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical form, it is possible to generate a set of equations defining P and Q on a less arbitrary basis. In the FPO formalism, the closed system becomes open because of an existing coupling between the localized closed system and the extended modes, i.e., because of the coupling between the Q and P subspaces. Due to this coupling, some discrete states of the closed system become resonance states of the open system which, in general, have finite lifetimes. In this work, expressions for scattering amplitudes are obtained for isolated as well as overlapping resonances. They relate the resonance properties to the properties of the Phononic structure and of the incident wave.


Fano-Feshbach Formalism for Elastic Waves

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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