Localized Floquet modes in arrays of out-of-phase curved waveguides with a quasiperiodic modulation

TL;DR

The paper addresses light localization in a photonic lattice formed by out-of-phase curved waveguides with an additional Aubry-André–type transverse modulation. It computes Floquet propagation constants by building a monodromy matrix over one longitudinal period, identifying pseudocollapses where Floquet-band widths shrink, and then shows that localized Floquet modes appear near those pseudocollapses and persist over continuous ranges of the modulation depth $\delta$ and drive amplitude $r$. The key finding is that localization can be achieved transversely while maintaining longitudinal periodic self-imaging, even when the corresponding straight-waveguide array is below the localization threshold, and that stronger $\delta$ expands the regime of robust localization. This work advances Floquet engineering in photonic lattices and suggests routes to robust diffraction inhibition, with potential extensions to nonlinear regimes and soliton-like localized Floquet modes.

Abstract

We study light propagation in an array of periodically curved waveguides consisting of pairs of waveguides with out-of-phase oscillations of waveguide centers. We compute the corresponding Floquet propagation constants and find pseudocollapses where the Floquet bands shrink and, respectively, light diffraction is significantly inhibited. When, in addition, the refractive index of the waveguides in the array have quasiperiodic modulation in the transverse direction, we establish the existence of Floquet modes localized in the transverse direction and periodic in the longitudinal direction. With increase of the depth of quasiperiodic modulation of the refractive index in the array, the localized Floquet modes emerge near the pseudocollapse points of the periodic array. In array with sufficiently high frequencies of waveguide oscillations, the localized Floquet modes can exist even for weak quasiperiodic modulation which is situated below the localization transition in the array of straight waveguides.

Figures (8)

• Figure 1: Schematics of an array with out-of-phase oscillating waveguides with slightly different refractive indices (which are illustrated by different thicknesses of the waveguides) due to imposed transverse quasiperiodic modulation of the refractive index. Indices $j$, $j+1$, etc. enumerate the waveguides; $x$- and $z$-axes correspond to the transverse and longitudinal directions, respectively. Two longitudinal periods of the structure are shown.
• Figure 2: Form-factors $\chi^{(2)}$ of guided modes of the array of straight waveguides with $r=0$, for different strengths of the quasiperiodic modulation $\delta$. The plot includes form-factors for $160$ modes with the largest propagation constants. In this figure and below, we use average waveguide depth $p^0=4.5$, the rational approximant $\varphi \approx 233/144$ of true irrational number $\varphi = (5^{1/2}+1)/2$, and phase shift $\theta \approx 1.1$.
• Figure 3: Real parts of Floquet propagation constants of eigenmodes of oscillating array for zero quasiperiodic modulation ($\delta=0$) and for increasing amplitude of longitudinal oscillations $r$ at $\omega=0.05$ (a) and $\omega=0.2$ (b). Each panel shows two replicas of the Floquet spectrum (two longitudinal Brillouin zones). Panel (c) shows the amplitudes $r^*$ at which Floquet band achieves a pseudocollapse (i.e. its width becomes minimal). The amplitudes corresponding to first four pseudocollapses are shown. Circles are obtained from the calculation, and solid lines are guides for an eye. Colors of the circles indicate the width of the band in each pseudocollapse point.
• Figure 4: Propagation of an input field corresponding to the single-waveguide excitation for amplitude of oscillations $r$ corresponding to three pseudocollapses at $\omega=0.05$ [$r=0.395, 0.617, 0.819$ in (a)-(c)] and one pseudocollapse at $\omega=0.2$ [$r=0.817$ in (d)]. Only a small part of the waveguide array is shown in each figure. The propagation distance corresponds to ten longitudinal periods $Z$ in each figure. The colorbar shows $|\Psi|$ and applies to all plots.
• Figure 5: Real parts of Floquet propagation constants $\beta$ and averaged form-factors $\tilde{\chi}^{(2)}$ versus amplitude of waveguide oscillations $r$ for Floquet modes in arrays with two different frequencies of the longitudinal oscillations $\omega$ and two different strengths of the quasiperiodic modulation $\delta$. Specific values of $\omega$ and $\delta$ are placed next to the corresponding plots.