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arXiv · 2601.06482

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

Hui Quan·Wei Si·Kai Jiang
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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.

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Hui Quan, Wei Si, Kai Jiang. 2026-01-10. Spectral Distribution of one-dimensional Photonic Quasicrystals: The Role of Irrational Numbers. https://arxiv.org/abs/2601.06482

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