Observation of a Localization Transition in Quasiperiodic Photonic Lattices

We report the observation of the signature of a localization phase transition for light in one-dimensional quasiperiodic photonic lattices, by directly measuring wave transport inside the lattice. Below the predicted transition point an initially narrow wave packet expands as it propagates, while above the transition expansion is fully suppressed. In addition, we measure the effect of focusing nonlinear interaction on the propagation and find it increases the width of the localized wave packets.

Figure 1

(a) Calculated amplitude profile of the ground state of an incommensurate lattice (horizontal axis) versus the strength of the incommensurate modulation depth $\lambda$ (vertical axis). A sharp localization transition is observed at $\lambda =2C$. The wave function is extended for $\lambda /C<2$, and localized to a single site for $\lambda /C>2$. A similar transition occurs for all other eigenstates of the system. (b) The same calculation for a disordered lattice. Here the localization length goes continuously to zero. All horizontal cross sections are normalized to a unit maximum. (c) A quasiperiodic lattice is obtained by modulating the on-site eigenvalues of the lattice sites (bars, ${\beta }_n={\beta }_0+\lambda \cos (2\pi n\chi )$) at a frequency $\chi$ that is incommensurate with the frequency of the lattice (dashed line).

Figure 2

Measurements of a localization transition in incommensurate photonic lattices. (a) The expansion of a single-site initial wave packet in a perfectly ordered lattice ($\lambda =0$) after 6 mm of propagation (blue) and 21 mm (red). (b) The same for an incommensurate lattice with $\lambda /C=1.6$. The wave packet still exhibits expansion, but at a reduced speed. (c) The same for an incommensurate lattice with $\lambda /C=3.1$. The wave packet remains tightly localized about the input position throughout the propagation, signifying localization. (d) Logarithmic plots of the averaged width of the wave packets (measured by the participation number PR) at different lengths and for different $\lambda$ values. These curves show the transition from expansion to localization as a function of the incommensurate modulation strength, signifying the localization transition.

Figure 3

(a) Comparison between the measured width of the wave packets after 21 mm of propagation (circles) and the theoretical values expected from Eq. (2) with $\gamma =0$ (line), for different values of $\lambda$. (b) Theoretical curves for the average width of the expanding wave packets as a function of the modulation ratio, for different propagation times. Each curve is normalized between 0 and 1. The curves show the transition becomes sharper for longer propagation times and is almost fully converged at $Cz=5$, corresponding to the experiment which is given on the same scale in blue circles.

Figure 4

Measurements of the effect of nonlinearity on wave packet expansion in incommensurate lattices. (a) The averaged participation ratios measured after 21 mm of propagation, as a function of power and for different incommensurate modulation depths $\lambda$. For $\lambda /C=0$, increased nonlinearity results in a reduced expansion of the wave packet. For $\lambda /C>2$ (the localized regime) nonlinearity results in weak expansion. (b) Measured output intensity distribution (horizontal axis) versus power for a single (nonaveraged) initial condition, in a lattice with $\lambda /C=3.1$. In the linear case the output is tightly localized to the input site, yet it spreads as the power is increased.