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Topological interface modes in systems with damping

Konstantinos Alexopoulos, Bryn Davies, Erik Orvehed Hiltunen

Abstract

We extend the theory of topological localised interface modes to systems with damping. The spectral problem is formulated as a root-finding problem for the interface impedance function and Rouché's theorem is used to track the zeros when damping is introduced. We show that the localised eigenfrequencies, corresponding to interface modes, remain for non-zero dampings. Using the transfer matrix method, we explicitly characterise the decay rate of the interface mode.

Topological interface modes in systems with damping

Abstract

We extend the theory of topological localised interface modes to systems with damping. The spectral problem is formulated as a root-finding problem for the interface impedance function and Rouché's theorem is used to track the zeros when damping is introduced. We show that the localised eigenfrequencies, corresponding to interface modes, remain for non-zero dampings. Using the transfer matrix method, we explicitly characterise the decay rate of the interface mode.

Paper Structure

This paper contains 24 sections, 15 theorems, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Mathematical setting
  4. Damped systems
  5. Differential problem
  6. Preliminaries
  7. Impedance functions
  8. Bulk index
  9. Undamped systems
  10. Real frequencies
  11. Complex frequencies
  12. Damped systems
  13. Asymptotic behaviour
  14. Transfer matrix formulation
  15. Asymptotic behaviour of damped systems
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $\mathfrak{A}(\delta)$ denote a common band gap of def:L:summary-eq:FB:summary. For each root $\omega _U$ of $Z^{(U)}$ in $\mathfrak{A}(0)$, there exists a root $\omega _D$ of $Z^{(D)}$ in $\mathfrak{A}(\delta)$, for $0<\delta\ll 1$, converging to $\omega _U$ as $\delta\to 0$.

Figures (4)

  • Figure 1: Numerically computed regions $R_c$ and $R_w$, where the inequality at the heart of Rouché's theorem does and does not apply, respectively. The roots of $Z^{(D_1)}$ and $Z^{(D_2)}$ are shown. The root $\omega _1$ of $Z^{(D_1)}$ (which has the smaller damping) must always lie on the interface between the two sets (i.e.$\omega _1\in \partial R_w$), while $\omega _2\in R_w$. In these cases, both $\omega _1$ and $\omega _2$ can be enclosed by a closed curve entirely lying in $R_c$, meaning that Rouché's theorem can be applied.
  • Figure 2: An example of a damped system. In the unit cell of each material, we have 3 particles. We notice the mirror-symmetric way in which the particles are placed inside the periodic cells. Each material is constituted by a semi-infinite array created by periodically repeating the unit cell. Our structure is the result of gluing materials $A$ and $B$ at the interface $x_0$.
  • Figure 3: Spectral bands of the damped system studied in Section \ref{['sec:Numerical example']}. We observe how the increase in damping pushes the bands further away from the real axis. Also, we notice how the position of the roots of the interface impedance function $Z^{(D)}$ in a spectral gap changes as the damping changes. For three different and fixed values of damping, we provide the exact structure of the spectral bands.
  • Figure 4: For fixed damping, we see the behaviour of an interface localised mode as $|x|\rightarrow\infty$, for the damped system considered in Section \ref{['sec:Numerical example']}. The eigenvalue envelope shows the decay rates in terms of the eigenvalues of the transfer matrices, as described in Theorem \ref{['cor:asymptotic behaviour']}.

Theorems & Definitions (26)

  • Theorem 2.1
  • Definition 4.1: Interface impedance
  • Lemma 4.2
  • proof
  • Definition 4.3: Bulk index
  • Theorem 5.1
  • Theorem 5.2
  • proof
  • Lemma 5.3
  • proof
  • ...and 16 more