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 the 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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• Photonics
• Quasicrystals