Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials

Ammari, Habib; Hiltunen, Erik Orvehed; Kosche, Thea

doi:10.1016/j.jde.2022.02.047

Cited by 12 publications

(22 citation statements)

References 29 publications

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“…In [14], it is shown that if A(t) is an analytical function of some parameter ε at ε = 0 and has the following expansion:…”

Section: Floquet-bloch Theory and Asymptotic Floquet Matrix Elementsmentioning

confidence: 99%

“…This characterization provides both theoretical insight and an efficient numerical method to compute the dispersion relationship of time-modulated systems of subwavelength resonators. A study of exceptional points in the case of time-modulated metamaterials is presented in [14]. Furthermore, in [1] the possibility of achieving non-reciprocal wave propagation in time-modulated metamaterials is shown.…”

Section: Introductionmentioning

confidence: 99%

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Unidirectional edge modes in time-modulated metamaterials

Ammari¹,

Jinghao²

2022

Preprint

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We prove the possibility of achieving unidirectional edge modes in time-modulated supercell structures. Such finite structures consist of two trimers repeated periodically. Because of their symmetry, they admit degenerate edge eigenspaces. When the trimers are time-modulated with two opposite orientations, the degenerate eigenspace splits into two one-dimensional eigenspaces described by an analytical formula, each corresponds to a mode which is localized at one edge of the structure. Our results on the localization and stability of these edge modes with respect to fluctuations in the time-modulation amplitude are illustrated by several numerical simulations.

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“…In [14], it is shown that if A(t) is an analytical function of some parameter ε at ε = 0 and has the following expansion:…”

Section: Floquet-bloch Theory and Asymptotic Floquet Matrix Elementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unidirectional edge modes in time-modulated metamaterials

Ammari¹,

Jinghao²

2022

Preprint

Self Cite

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“…where ε > 0 is some small parameter describing the amplitude of the time modulation and M α 0 corresponds to the unmodulated case. We can assume that the above series converges for |ε| < ε 0 , where ε 0 > 0 is independent of t, provided that the modulations of ρ and κ have finitely many non-zero Fourier coefficients [19,21].…”

Section: Problem Formulationmentioning

confidence: 99%

“…We shall make use of the asymptotic Floquet analysis developed in [21], which is a combination of perturbation analysis and Floquet theory; see also [43]. We can rewrite the second-order ODE ( 16) into…”

Section: Problem Formulationmentioning

confidence: 99%

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Transmission properties of space-time modulated metamaterials

Ammari¹,

Jinghao²,

Zeng³

2022

Preprint

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We prove the possibility of achieving exponentially growing wave propagation in spacetime modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k-gaps in the band structures of space-time modulated systems of subwavelength resonators.

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Topological phenomena in honeycomb Floquet metamaterials

Ammari

Kosche

2023

Math. Ann.

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Being driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations $$\left\{ \frac{d}{dt}X = A_\alpha (t)X\right\} _{\alpha \in {\mathbb {T}}^d}$$ d dt X = A α ( t ) X α ∈ T d . In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence of solid-state physics is possible in the case of Floquet metamaterials, i.e., subwavelength high-contrast time-modulated metamaterials. This paper is a first step in that direction. Since the bulk-boundary correspondence states that topological properties of the bulk materials characterize the occurrence of edge modes, we dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should thus be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation $$\left\{ \frac{d}{dt}X = A_\alpha (t)X\right\} _{\alpha \in {\mathbb {T}}^d}$$ d dt X = A α ( t ) X α ∈ T d , we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form F, P of the associated fundamental solution $$\left\{ X_\alpha (t) = P(\alpha ,t)\exp (tF_\alpha )\right\} _{\alpha \in {\mathbb {T}}^d}$$ X α ( t ) = P ( α , t ) exp ( t F α ) α ∈ T d . This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix $$X_\alpha (T)$$ X α ( T ) and the Lyapunov transformation $$P(\alpha ,t)$$ P ( α , t ) . The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of a hexagonal structure. We provide two interesting examples of topologically non-trivial time-modulated hexagonal structures.

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Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials

Cited by 12 publications

References 29 publications

Unidirectional edge modes in time-modulated metamaterials

Unidirectional edge modes in time-modulated metamaterials

Transmission properties of space-time modulated metamaterials

Topological phenomena in honeycomb Floquet metamaterials

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