Dynamic resonance in the high-Q and near-monochromatic regime

Abeynanda, Gayan S.; Shipman, Stephen P.

doi:10.1109/mmet.2016.7544100

Cited by 10 publications

(14 citation statements)

References 14 publications

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“…3.5 p.160]) which is possibly non uniquely defined if there is a trapped mode 1 at the wavenumber k. However, the reflection coefficient R ∈ C and transmission coefficient T ∈ C are always uniquely defined. They satisfy the energy conservation relation |R| 2 + |T| 2 = 1 already written in (1). Of course R and T depend on the features of the geometry, in particular on L. In this work, we explain how to find some L such that R = 0, |T| = 1 (non reflectivity); |R| = 1, T = 0 (perfect reflectivity); or R = 0, T = 1 (perfect invisibility).…”

Section: Introductionmentioning

confidence: 94%

Invisibility and Perfect Reflectivity in Waveguides with Finite Length Branches

Chesnel¹,

Назаров²,

Pagneux³

2018

SIAM J. Appl. Math.

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We consider a time-harmonic wave problem, appearing for example in water-waves theory, in acoustics or in electromagnetism, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients R, T as the length of the branch increases and we show how to design geometries where non reflectivity (R = 0, |T| = 1), perfect reflectivity (|R| = 1, T = 0) or perfect invisibility (R = 0, T = 1) hold. Numerical experiments illustrate the different results.

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Section: Introductionmentioning

confidence: 94%

Invisibility and Perfect Reflectivity in Waveguides with Finite Length Branches

Chesnel¹,

Назаров²,

Pagneux³

2018

SIAM J. Appl. Math.

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“…The main part of this work is dedicated to prove rigorously that the scattering matrix is not smooth at (ε, λ) = (0, λ 0 ). Our study shares similarities with [45,44,46,1]. In these articles, the authors analyse the Fano resonance phenomenon in gratings in electromagnetism, a context close to ours, via techniques of complex analysis and by means of tools of the theory of analytic functions of several variables.…”

Section: Introductionmentioning

confidence: 54%

Non reflection and perfect reflection via Fano resonance in waveguides

Chesnel

Назаров

2018

Communications in Mathematical Sciences

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We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter ε and of the frequency λ is in general not continuous at a point (ε, λ) = (0, λ 0 ) where trapped modes exist. In particular, we prove that for a given ε = 0 small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of λ 0 . We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.

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“…(6) As noted in §1.1, when the obstacle O contains an ellipse-shaped cavity, the resolvent grows exponentially through a sequence k j (1.7); in this situation Theorem 1.1 implicitly contains information about the widths of the peaks in the norm of the resolvent at k j . We are not aware of any results in the literature about the widths of these peaks in the setting of obstacle scattering, but precise information about the widths and heights of peaks in the transmission coefficient for model resonance problems in one space dimension can be found in [1,105]. (7) Complementary results (in a different direction to Theorem 1.1) about "good" behaviour of the resolvent in trapping scenarios can be found in in [24, theorem 1.1], [17, theorem 4], and [39, theorems 1.1, 1.2].…”

Section: Statement Of Main Results (In the Setting Of Impenetrable-dirichlet-obstacle Scattering)mentioning

confidence: 99%

For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

Lafontaine

Spence

Wunsch

2020

Comm Pure Appl Math

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It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial‐growth assumption on the solution operator (e.g. hp‐finite elements, hp‐boundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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Dynamic resonance in the high-Q and near-monochromatic regime

Cited by 10 publications

References 14 publications

Invisibility and Perfect Reflectivity in Waveguides with Finite Length Branches

Invisibility and Perfect Reflectivity in Waveguides with Finite Length Branches

Non reflection and perfect reflection via Fano resonance in waveguides

For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

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