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Universal estimates for the density of states for aperiodic block subwavelength resonator systems

Habib Ammari, Silvio Barandun, Bryn Davies, Erik Orvehed Hiltunen, Alexander Uhlmann

TL;DR

The paper develops a universal framework to estimate the density of states for 1D aperiodic block subwavelength resonator systems. By proving DoS convergence to a deterministic limit and revealing a tripartite spectral structure—bandgaps with zero DoS, smooth shared pass bands, and fractal-like hybridisation regions—the authors enable efficient prediction through a meta-atom approach that leverages local block arrangements. This method remains effective under various sampling schemes, including quasiperiodic and hyperuniform, and provides practical tools for rapid DoS estimation in designed metamaterials. The work combines transfer/propagation matrices, Jacobi operator theory, and ergodic arguments to yield both rigorous results and computationally scalable techniques with potential impact on subwavelength device design.

Abstract

We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks $M\to \infty$, the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

Universal estimates for the density of states for aperiodic block subwavelength resonator systems

TL;DR

The paper develops a universal framework to estimate the density of states for 1D aperiodic block subwavelength resonator systems. By proving DoS convergence to a deterministic limit and revealing a tripartite spectral structure—bandgaps with zero DoS, smooth shared pass bands, and fractal-like hybridisation regions—the authors enable efficient prediction through a meta-atom approach that leverages local block arrangements. This method remains effective under various sampling schemes, including quasiperiodic and hyperuniform, and provides practical tools for rapid DoS estimation in designed metamaterials. The work combines transfer/propagation matrices, Jacobi operator theory, and ergodic arguments to yield both rigorous results and computationally scalable techniques with potential impact on subwavelength device design.

Abstract

We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks , the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

Paper Structure

This paper contains 20 sections, 7 theorems, 50 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and tools
  3. Subwavelength resonators
  4. Block disordered systems
  5. Sampling
  6. Propagation matrix formalism
  7. Jacobi operators and metric transitivity
  8. Metric transitivity
  9. Consequences
  10. Fractal density of states in the hybridisation region
  11. Perturbative characterisation
  12. Meta-atom construction
  13. Variable decay
  14. Dependent sampling
  15. Bound length sampling
  16. ...and 5 more sections

Key Result

Theorem 2.1

Consider a system consisting of $N$ subwavelength resonators in $\mathbb{R}$. Then, there exist exactly $N$ subwavelength resonant frequencies $\omega(\delta)$ that satisfy $\omega(\delta)\to 0$ as $\delta\to 0$. Moreover, the $N$ resonant frequencies are given by where $(\lambda_i)_{1\leq i\leq N}$ are the eigenvalues of the eigenvalue problem Furthermore, the corresponding resonant modes $u_i(

Figures (9)

  • Figure 1: A block disordered system consisting of two single resonator blocks $B_1$ and a dimer block $B_2$ arranged in a chain given by the sequence $\chi= (1,2,1)$. It thus consists of $M=3$ blocks and $N=4$ resonators $D_1,\dots ,D_4$ in total.
  • Figure 2: Left: Comparison of maximal block propagation matrix eigenvalue $\left\lvert\xi_2(\lambda)\right\rvert$ for both blocks with Thouless ratios $g(\lambda_i)$ of the entire random block disordered system consisting of these block (each colored vertical line corresponds to the Thouless ratio $g(\lambda_i)$ of the eigenvalue $\lambda_i$). Where there are no such lines, the density of eigenvalues is zero. Right: Cumulative density function (CDF) of the total system, laid atop the different spectral regions. We can see that the CDF arrange smoothly in the shared pass band and is constant in the bandgap, reminiscent of the periodic case. However, in contrast to the periodic case, we observe an additional hybridisation region where the CDF grows fractal-like.
  • Figure 3: Convergence of empirical cumulative density functions under increasing system size. We consider random block disordered systems consisting of single resonator and dimer blocks as in \ref{['ex:standard_blocks']} sampled with either equal density ($p_{single}=p_{dimer}=1/2$) or low dimer density ($p_{dimer}=1/10$). We calculate the eCDF for large arrays ($M=2^{13}$) and compare this to the eCDF of smaller resonator arrays ($M=2^p, p=2,\dots 11$) averaged over $M=10^5$ realisations.
  • Figure 4: Densities of states for block disordered systems ($M = 10^5$) with varying dimer density, together with various dimer defect modes. We can see that in both cases (but especially in the low dimer density case) the peaks in the density of states closely correspond to the defect modes obtained by considering various local arrangements of dimers. Both systems consist of single resonator and dimer blocks as described in \ref{['ex:standard_blocks']}. Left: Random sampling with equal probability $p_{single}=p_{dimer} = \frac{1}{2}$. Right: Random sampling with lower dimer density $p_{dimer}=\frac{1}{10}$.
  • Figure 5: Convergence of the meta-atom estimate using \ref{['alg:upper_spectrum']} to the empirical CDF of a large system as the meta-atom length $L$ and single resonator amount $P$ is increased. Already for small numbers of $L$ and $P$, the estimate agrees extremely well with the empirical CDF.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Theorem 2.1
  • Example 2.2
  • Definition 2.3: Independently and identically sampled discrete sequences
  • Example 2.4
  • Definition 2.5: Propagation matrices
  • Theorem 2.6
  • Definition 3.1: Metrically transitive group
  • Lemma 3.2
  • proof
  • Definition 3.3: Jacobi Operator
  • ...and 10 more