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Transmission properties of time-dependent one-dimensional metamaterials

Habib Ammari, Jinghao Cao, Erik Orvehed Hiltunen, Liora Rueff

TL;DR

This work analyzes wave propagation in a one-dimensional chain of high-contrast subwavelength resonators with periodically time-modulated material parameters under quasi-periodic boundary conditions. The authors solve the time-harmonic problem exactly using a Dirichlet-to-Neumann map and compute subwavelength quasifrequencies via Muller's method, then derive a capacitance-matrix approximation that yields an ODE for the quasifrequencies. Numerical comparisons show the capacitance approach is accurate and substantially more efficient than full Muller's method, while capturing key phenomena such as band gaps, k-gaps, and non-reciprocity induced by time modulation. The study demonstrates non-reciprocal wave transmission in the subwavelength regime and provides asymptotic analysis and practical guidance for parameter choices. The results bridge subwavelength metamaterials theory with one-dimensional effective models and offer reproducible code for further exploration.

Abstract

We solve the wave equation with periodically time-modulated material parameters in a one-dimensional high-contrast resonator structure in the subwavelength regime exactly, for which we compute the subwavelength quasifrequencies numerically using Muller's method. We prove a formula in the form of an ODE using a capacitance matrix approximation. Comparison of the exact results with the approximations reveals that the method of capacitance matrix approximation is accurate and significantly more efficient. We prove various transmission properties in the aforementioned structure and illustrate them with numerical simulations. In particular, we investigate the effect of time-modulated material parameters on the formation of degenerate points, band gaps and k-gaps.

Transmission properties of time-dependent one-dimensional metamaterials

TL;DR

This work analyzes wave propagation in a one-dimensional chain of high-contrast subwavelength resonators with periodically time-modulated material parameters under quasi-periodic boundary conditions. The authors solve the time-harmonic problem exactly using a Dirichlet-to-Neumann map and compute subwavelength quasifrequencies via Muller's method, then derive a capacitance-matrix approximation that yields an ODE for the quasifrequencies. Numerical comparisons show the capacitance approach is accurate and substantially more efficient than full Muller's method, while capturing key phenomena such as band gaps, k-gaps, and non-reciprocity induced by time modulation. The study demonstrates non-reciprocal wave transmission in the subwavelength regime and provides asymptotic analysis and practical guidance for parameter choices. The results bridge subwavelength metamaterials theory with one-dimensional effective models and offer reproducible code for further exploration.

Abstract

We solve the wave equation with periodically time-modulated material parameters in a one-dimensional high-contrast resonator structure in the subwavelength regime exactly, for which we compute the subwavelength quasifrequencies numerically using Muller's method. We prove a formula in the form of an ODE using a capacitance matrix approximation. Comparison of the exact results with the approximations reveals that the method of capacitance matrix approximation is accurate and significantly more efficient. We prove various transmission properties in the aforementioned structure and illustrate them with numerical simulations. In particular, we investigate the effect of time-modulated material parameters on the formation of degenerate points, band gaps and k-gaps.
Paper Structure (16 sections, 12 theorems, 79 equations, 6 figures)

This paper contains 16 sections, 12 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminary theory
  3. Problem formulation
  4. Time-dependent material parameters
  5. Exact solution
  6. Exterior problem
  7. Interior problem
  8. Explicit choice of parameters
  9. Muller's method
  10. Capacitance approximation and asymptotic analysis
  11. Capacitance matrix formulation
  12. Asymptotic analysis
  13. Physical interpretation and numerical simulations
  14. Conclusion
  15. Code availability
  16. ...and 1 more sections

Key Result

Lemma 3.1

Assume that $k^n=(\omega+n\Omega)/v_0$, for some fixed $n\in\mathbb{Z}$, is not of the form $m\pi/\ell_{i(i+1)}$ for some non-zero integer $m\in\mathbb{Z}\backslash\{0\}$ and index $1\leq i\leq N$. Then, for any quasi-periodic sequence $(f_i^{\pm})_{1\leq i\leq N}\in\mathbb{C}^{2N,\alpha}$, there ex Furthermore, when $k^n\neq 0$, the solution $v_{f,n}^{\alpha}$ reads explicitly where, for fixed $

Figures (6)

  • Figure 1: An illustration of the one-dimensional setting for $N=3$ resonators in the unit cell.
  • Figure 2: The time it takes to formulate the root-finding problem and for Muller's method to solve it depending on the truncation parameter $K$. The runtime depends algebraically on the parameter $K$.
  • Figure 3: We compare the quasifrequencies obtained by Muller's method with the quasifrequencies obtained through the capacitance matrix approximation.
  • Figure 4: Comparing the runtime of the exact computation for $K=3$ and the capacitance approximation for the following parameter values: $\delta=0.0001,\,\Omega=0.05,\varepsilon_{\kappa}=\varepsilon_{\rho}=0.4,\,\phi_{i}=\pi/i,\,v_0=v_{\mathrm{r}}=1,\,\ell_i=\ell_{i(i+1)}=1$.
  • Figure 5: Subwavelength quasifrequencies for three resonators repeated periodically in the static case. The figures on the right-hand side illustrate the setting corresponding to the numerical results shown in the left-hand side figures.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Lemma 3.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.2
  • Lemma 3.3
  • Remark 3.3
  • Theorem 3.4
  • proof
  • ...and 17 more