Reentrant delocalization transition in one-dimensional photonic quasicrystals

Waves propagating in certain one-dimensional quasiperiodic lattices are known to exhibit a sharp localization transition. We theoretically predict and experimentally observe that the localization of light in one-dimensional photonic quasicrystals is followed by a second delocalization transition for some states on increasing quasiperiodic modulation strength—an example of a reentrant transition. We further propose that this phenomenon can be qualitatively captured by a dimerized tight-binding model with long-range couplings. This work furthers our understanding of localization physics in complex systems and the impact of quasiperiodicity on light transport in photonic devices.

Figure 1

Schematic of multilayer photonic structures made out of Si and ${\mathrm{SiO}}_2$ layers. The structures have increasing quasiperiodic modulation of layer thicknesses (from left to right); the leftmost structure is a perfect one-dimensional photonic crystal and the rest are photonic quasicrystals.

Figure 2

(a) Eigenvalue spectrum of the PhQC states and their corresponding IPRs as a function of $A$ for $N=89$ and $\beta =144/89$. (b) The transmission spectrum as a function of $A$ for $N=89$. Localization of various states corresponds to sharp drops in transmission (white arrows). Some states undergo a second delocalization transition around $A=0.8$, which results in a sharp recovery of transmission (blue arrow). (c) Normalized $\mathcal{H}\left(z\right)$-field profiles of a state near the black arrow in (a), for various values of $A$.

Figure 3

(a) Scanning electron microscope (SEM) image of a cut through a typical one-dimensional photonic quasicrystal fabricated by PECVD. Both layers in a pair of neighboring Si and ${\mathrm{SiO}}_2$ layers have identical thicknesses. The thickness values of each such pair are modulated according to Eq. (2). (b) Experimentally measured transmission spectrum as a function of $A$ for $N=13$. (c) Simulated transmission spectrum as a function of $A$ for $N=13$. In (b) and (c), the localization transitions are marked with white arrows and the reentrant delocalization transition is marked with a blue arrow.

Figure 4

(a) Schematic of the tight-binding model. The dimerized unit cell for the periodic system ($\alpha =0$) is highlighted. The solid (dotted) lines represent nearest-neighbor (next-nearest-neighbor) couplings. (b) The energy spectrum and IPR of the states of the model for $E_{i,A}=1, E_{i,B}=2, t_{\text{NN}}=0.7, t_{\text{NNN}}=0.35$, and $N=89$. The states of the second band exhibit a mobility edge and are localized for $0.25<\alpha <0.6$. Some states of this band undergo a reentrant delocalization transition at $\alpha \sim 0.6$. (c) A plot of the $\langle \text{IPR}\rangle$ and $\langle \text{NPR}\rangle$ for the states of the upper band for large $N$. The highlighted areas indicate intermediate regimes, where both $\langle \text{IPR}\rangle$ and $\langle \text{NPR}\rangle$ are nonzero, and localized and extended states coexist.

Figure 5

(a) Transmission spectrum for $N=13$. This is the system size fabricated in the experiment. (b) Transmission spectrum for $N=144$. (c) Transmission spectrum for $N=2584$. The reentrant transition occurs for arbitrarily large system sizes.

Figure 6

Transmission spectrum vs quasiperiodic modulation strength ($A$) for the 1D PhQCs with varying system sizes: (a) $N=8$, (b) $N=13$, (c) $N=21$, (d) $N=34$, (e) $N=55$, and (f) $N=89$. (g), (h) The $\mathcal{H}\left(z\right)$-field profiles of the states for $N=13$ marked with the blue and white arrows, respectively.

Figure 7

The eigenvalue spectrum for the family of PhQCs considered in the main text for a larger frequency range. In the main text and in the experiment, we focus on the localization features of states in the dimensionless frequency range of 0.227–0.357 (dashed box) that constitute the second band of the PhC limit (i.e., at $A=0$).

Figure 8

Comparison of a sample's layer thicknesses extracted from SEM images (blue) and targeted thickness (red).