Spectral Distribution of one-dimensional Photonic Quasicrystals: The Role of Irrational Numbers

TL;DR

This work tackles the spectral distribution of 1D photonic quasicrystals formed by two incommensurate harmonics with ratio $β$ by employing the projection method to embed the structure in a 2D periodic superspace and compute the spectrum at the Γ point. It uncovers a butterfly-shaped spectrum with abundant gaps and demonstrates a robust, magnitude-dependent linear relation for spectral statistics: in the strong quasiperiodic regime, the spectral-structure factor $Q$ satisfies $Q=1-β$ for $β<β_c$ and $Q=β$ for $β>β_c$ with $β_c≈0.424$, independent of the irrational type. The study also shows that localized states tend to appear at spectral-gap edges, and the number of eigenvalues within a finite bandwidth grows with numerical resolution $N$, highlighting the non-compact nature of the underlying operator. These findings advance understanding of how the irrational parameter $β$ governs the spectral properties of 1D PQCs and offer insights for tailoring band gaps in quasiperiodic photonic devices.

Abstract

In this paper, we construct a one-dimensional photonic quasicrystal by combining two incommensurate spatial harmonics, where the ratio of their periods is the irrational number β. We evaluate the photonic quasicrystal accurately by a generalized spectral method that embeds the quasiperiodic structure into a higher-dimensional periodic system. We study the spectral distribution of one-dimensional photonic quasicrystals and find some interesting phenomena. As the computational resolution N increases, there are more eigenvalues within finite frequency bandwidths, and the maximum localization always occurs at spectral gap edges for states near index N + 1. By varying βwithin the range of (0,1), we present a butterfly-shaped spectral structure with abundant band gaps. We find that the spectral structure factor Q (defined as I_{mg}/N, where I_{mg} is the maximum gap index) exhibits different linear patterns as βchanges: Q = 1 - βwhen β< βc, while Q = βwhen β> βc, where βc \approx 0.424 is the transition point. This linear relationship holds robustly in the strong quasiperiodic regime (βaway from 0 or 1) and is independent of the specific type of irrational number used. The relationship disappears (weak quasiperiodic regime) near β= 0 or β= 1. It demonstrates that the intrinsic spectral properties of one-dimensional photonic quasicrystals are fundamentally governed by the magnitude of the irrational parameter β.

Figures (5)

• Figure 1: (a)-(c) The eigenvalue spectral structure corresponding to the golden ratio $(\sqrt{5} - 1)/2$ at calculation resolutions of $N=150$, $200$ and $250$. As the resolution N increases, the number of spectra within a given frequency bandwidth also increases. (d) The field $H(x)$ distributions of the three eigenstates at a resolution of $N = 250$ in physical space $X$, which exhibit extended, localized, and extended characteristics, respectively.
• Figure 2: (a) Index of eigenvalue spectra versus irrational number $\beta$, where color represents the value of eigenvalue. (b) The IPR distribution corresponding to (a). The maximum spectral gaps correspond to the bright IPR stripes, marked by arrows and green dashed lines. (c)(d) The field amplitude $|H(x)|$ distribution of the 11 eigenstates marked by green arrows in (b), which gradually revert to the extended states as the index increases.
• Figure 3: (a) The butterfly-shaped spectral structure with a calculation resolution of $N = 250$, where the first $N$ eigenvalue spectra are marked in dark red. (b) The maximum gap distribution is extracted from the first $N$ eigenvalue spectra at different computational resolutions. (c) The index of the maximum gap ($I_{mg}$) extracted at different resolutions $N$. The interval (0,1) is divided into regions I, II, III, and IV, which are marked with light orange (strong quasiperiodic) and light blue (weak quasiperiodic). (d) The factor $Q$ ($Q = I_{mg}/N$) maintains a linear relationship with $\beta$ in the strong quasiperiodic region, while it disappears in the weak quasiperiodic region. This phenomenon is independent of the type of irrational number.
• Figure 4: The lattice potentials (blue curves) and eigenvalue spectra (dark red scattered points) corresponding to the irrational numbers in regions I–IV. (a)(b) For the irrational numbers in regions I and IV (close to 0 and 1, respectively), the envelope of the lattice potential $\varepsilon^{-1}(x)$ exhibits a periodic-like feature, and the eigenvalue spectra contain many short and small gaps. (c)(d) For irrational numbers in regions II and III (far from 0 and 1), the lattice potential $\varepsilon^{-1}(x)$ is the strong quasiperiodic, and the eigenvalue spectra show obvious gaps. The $I_{mg}$ is marked by the green arrow.
• Figure 5: (a) Evolution of the first two maximum eigenvalue spectra gaps (Gap1, Gap2) for irrational numbers near the critical point $\beta_c$ of regions II and III. (b) The irrational number to the left of the critical point ($\beta<\beta_c$) satisfies Gap1$<$Gap2. (c) The irrational number to the right of the critical point ($\beta>\beta_c$) satisfies Gap1$>$Gap2. The indices of Gap1 and Gap2 are marked by green arrows.