Energy estimates for wave amplification in quasiperiodic Fibonacci time-modulated media

Abstract

Fibonacci time quasicrystals can be approximated by temporal supercells to reveal a fractal collection of $k$ gaps, in which wave energy is amplified exponentially. These estimates are validated by the observation of "super" $k$ gaps that are independent of the duration of the temporal supercell. This approach predicts the regions of parametric amplification and provides accurate estimates of the energy growth rate.

Figures (4)

• Figure 1: Piecewise-constant modulation according to a Fibonacci rule. (a) The coefficient $\Omega(t)$ alternates between $\Omega_0$ and either $\Omega_A$ or $\Omega_B$, depending on the corresponding letter in the Fibonacci word. (b) For certain parameter values (here, $k=3.2$) the solution grows exponentially. (c) The (exponential) rate of energy growth can be estimated using a supercell approximation.
• Figure 2: Dispersion curves for the temporal supercells given by the Fibonacci words (a) $\mathcal{F}_3=ABA$ and (b) $\mathcal{F}_5=ABAABABA$. The $k$ gaps are shaded above each plot.
• Figure 3: Supercell approximations can be used to predict stability. Here, (a) The pattern of $k$ gaps for successive supercell approximations. The $k$ gaps are shaded for successive Fibonacci unit cells, indexed by $n$. (b) The growth rate, estimated using the $\mathcal{F}_5$ supercell. The energy $E(t)$ for (c) $k=1.25$ and (d) $k=2.13$. In both cases, the data are plotted alongside the slope predicted by the $\mathcal{F}_5$ supercell approximation.
• Figure 4: The supercell estimates for the energy growth rate converge as the duration of the unit cell is increased. Here, $k=3.2$ is in a super $k$ gap (see Fig. \ref{['fig2']}(a)). The growth rate is estimated as $E(t)=\exp(r_n t)$ where the growth rate $r_n$ is based on the $\mathcal{F}_n$ supercell. The convergence of the growth rates $r_n$ is illustrated inset.