Hierarchical Band Gaps in Complex Periodic Systems

TL;DR

The paper addresses predicting band gaps in complex one-dimensional periodic systems by decomposing a large unit cell into simpler constituents and analyzing their transfer matrices. It proves that for matrices $T_i=\begin{pmatrix}0&-1\\1&t_i\end{pmatrix}$ with $|t_i|>2$, any finite product remains in the band-gap set ${\mathcal M}=\{M\in SL_2(\mathbb{R}) : |\mathrm{tr}(M)|>2\}$, thereby guaranteeing hierarchical band gaps when constituents share a common gap. This provides a practical, sufficient criterion to predict gaps in a variety of 1D systems, including masses-springs, coupled pendulums, resonant phononic crystals, and Fibonacci tilings, without full spectral computation. The framework also discusses limitations (sufficiency, not necessity) and outlines directions for generalization to modulated couplings and higher dimensions.

Abstract

Complex periodic structures inherit spectral properties from the constituent parts of their unit cells, chiefly their spectral band gaps. Exploiting this intuitive principle, which is made precise in this work, means spectral features of periodic systems with very large unit cells can be predicted without numerical simulation. We study a class of difference equations with periodic coefficients and show that they inherit spectral gaps from their constituent elements. This result shows that if a frequency falls in a band gap for every constituent element then it must be in a band gap for the combined complex periodic structure. This theory and its instantaneous utility is demonstrated in a series of vibro-acoustic and mechanical examples.

Figures (9)

• Figure 1: This work develops a theory for hierarchical band gaps in complex periodic systems. These are band gaps in complex periodic systems which are common with the band gaps of all the constituent simple systems. For example, the spectra of three materials A, B and C are shown here, with their common band gaps highlighted and inherited by the combined periodic material with unit cell ABC. This presents a simple way to predict some, but not all, of the band gaps in complex periodic systems.
• Figure 2: The canonical difference equation studied here can be used to model a system of masses (with modulated masses $m_i$) coupled by identical springs (with spring constant $\kappa$).
• Figure 3: Hierarchical band gaps of a system of coupled masses and springs, where the complex system inherits the common high-frequency band gap of its constituent elements. A simple periodic structure is considered, composed of five constituent elements.
• Figure 4: A system of coupled pendulums (which are coupled by indentical springs) can be described by difference equations of the form considered in this work.
• Figure 5: Hierarchical band gaps of a system of coupled pendulums, with a simple periodic unit cell consisting of five elements. The complex pendulum periodic structure inherits both high- and low-frequency hierarchical band gaps from the constituent systems.

Theorems & Definitions (9)

• Definition 2.1
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.1
• proof
• Theorem 3.1
• proof