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

TL;DR

This study investigates the spectral properties of one-dimensional photonic quasicrystals constructed with irrational ratios, revealing a linear relationship in spectral structure factor and the influence of irrational parameters on spectral features.

Contribution

It introduces a generalized spectral method for accurately analyzing quasiperiodic structures and uncovers fundamental spectral relationships governed by the irrational parameter eta.

Findings

Spectral distribution exhibits more eigenvalues with increasing resolution.

Maximum localization occurs at spectral gap edges near index N+1.

Spectral structure factor Q shows a linear dependence on eta, with a transition point at eta c pprox 0.424.

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 \beta. 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 \beta within 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 \beta changes: Q = 1 - \beta when \beta < \beta c, while Q = \beta when \beta > \beta c, where \beta c \approx 0.424 is the transition point. This linear relationship holds robustly in the strong quasiperiodic regime (\beta away from 0 or 1) and is independent of the specific type of irrational number used. The relationship disappears (weak quasiperiodic regime) near \beta = 0 or \beta = 1. It demonstrates that the intrinsic spectral properties of one-dimensional photonic quasicrystals are fundamentally governed by the magnitude of the irrational parameter \beta.