Quasinormal mode representation of radiating resonators in open phononic systems

Vincent Laude and Yan-Feng Wang

Phys. Rev. B 107, 144301 – Published 4 April 2023

Abstract

Open phononic systems including resonators radiating inside an unbounded medium support localized phonons characterized by a complex frequency. In this context, the concept of elastic quasinormal mode (QNM) arises naturally, as in the cases of nanophotonic and plasmonic open systems. Based on a complex, unconjugated form of reciprocity theorem for elastodynamics, the eigenfunction expansion theorem expressed on the elastic QNM basis yields an accurate approximation to the response function, for an arbitrary excitation. The description of the elastic Purcell effect then requires defining a complex-valued modal volume for each QNM. For validation, we first consider the case a vibrating nylon rod radiating in water. As a second test example, we consider a slender nickel ridge on the surface of a fused silica substrate, before extending our attention to a nanoscale tuning fork composed of two such ridges. In all cases, the response estimated from only a few elastic QNMs agrees with the solution to the elastodynamic equation.

Received 28 November 2022
Revised 21 February 2023
Accepted 28 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.144301

Physics Subject Headings (PhySH)

Continuum mechanics Elastic waves Elasticity

Finite-element method Integral equations

General PhysicsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Vincent Laude

Université de Franche-Comté, CNRS, institut FEMTO-ST, F-25000 Besançon, France

Yan-Feng Wang

Department of Mechanics, School of Mechanical Engineering, Tianjin University, 300350 Tianjin, China

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 14 — 1 April 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Definition of supporting domains for wave resonance and propagation. (a) Finite, closed domain supports normal modes. (b) Infinite, open domain supports quasinormal modes. (c) These can be approximated by closing the domain of computation with a perfectly matched layer (PML), that is the truncated image of the infinite domain in (b) in a complex coordinate transformation.
Reuse & Permissions
Figure 2
Stochastic response for a solid nylon rod with elliptical cross-section vibrating in water, as a function of the reduced frequency $ω d / (2 π)$ . $d$ is the diameter of the rod, or length of the long axis of the ellipse. The short axis length is $0.8 d$ . The exact forced response is plotted with the magenta line, whereas the eigenexpansion solution is plotted with the green line. There are five damped resonances in the frequency range of interest, labeled from 0 to 4. The corresponding real parts of the pressure of QNMs are shown, with the color bar going from blue (negative values) to red (positive values). The characteristics of QNMs are listed in Table 1.
Reuse & Permissions
Figure 3
A nickel ridge on a fused silica substrate. The ridge is 100- $n m$ wide and 1000- $n m$ high. The frequency response is obtained for a body force on the ridge applied along the $x$ axis only ( $F_{x} = 1$ ). The result of the superposition of the four quasinormal modes identified in the frequency range of interest is plotted on top of the frequency response computed as a function of frequency by solving the forced elastodynamic equation. The modulus of the total displacement for the four QNMs is plotted, together with the deformed mesh in-plane. The colorbar scales from black (zero total displacement) to green (maximum total displacement).
Reuse & Permissions
Figure 4
A pair of nickel ridges on a fused silica substrate. The ridges are $w = 100$ $n m$ wide and $h = 1000$ $n m$ high, and are separated by $δ = 150$ $n m$ from center to center. The frequency response is obtained for a body force on the left ridge only, applied along the $x$ axis only ( $F_{x} = 1$ ). The result of the superposition of the eight quasinormal modes identified in the frequency range of interest is plotted on top of the frequency response computed as a function of frequency by solving the forced elastodynamic equation. The modulus of the total displacement for four of the QNMs is plotted, together with the deformed mesh in-plane. The colorbar scales from black (zero total displacement) to green (maximum total displacement).
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics