Observation of quincunx-shaped and dipole-like flatband states in photonic rhombic lattices without band-touching

Abstract

We demonstrate experimentally the existence of compact localized states (CLSs) in a quasi-one-dimensional photonic rhombic lattice in presence of two distinct refractive-index gradients (i.e., a driven lattice ribbon) acting as external direct current (dc) electric fields. Such a lattice is composed of an array of periodically arranged evanescently coupled waveguides, which hosts a perfect flatband that touches both remaining dispersive bands when it is not driven. The external driving is realized by modulating the relative writing beam intensity of adjacent waveguides. We find that a y-gradient set perpendicularly to the ribbon preserves the flatband while removing the band-touching. The undriven CLS -which occupies two lattices sites over one unit cell -turns into a quincunx-shaped CLS spanned over two unit cells. Instead, an x-gradient acting parallel to the ribbon yields a Stark ladder of CLS whose spatial profile is unchanged with respect to the undriven case. We notably find that their superposition leads to Bloch-like oscillations in momentum space.

Photonic flat-band lattices and unconventional light localization

“…where [81] lattices with different boundary terminations, (C-E) Kagome lattices with different boundary terminations [86], (F) a superhoneycomb lattice [85], and (G) a "driven" rhombic lattice with an index gradient [84].…”

“…Flat-band states have also been demonstrated in a rhombic lattice that is robust in the presence of an external driving field (index gradient) along the lattice axis by Mukherjee and Thomson [38]. Recently, Xia et al [84] demonstrated the existence of CLSs in a driven rhombic lattice ribbon with two distinct refractive index gradients acting as external dc fields. Such CLSs span over two unit cells with a quincunx shape under the action of a driving force.…”

“…Like graphene, different edges support different edge states and have different characteristics. For instance, some flat-band states can only exist in lattices with appropriate boundaries, such as the extended flat-band line states [81,[84][85][86] discussed in later sections. In order to observe such states, we develop a simple cw-laser-writing technique which can establish finite-sized photonic lattices with desired boundaries.…”

“…These works attracted wide interest because they prove that waves can stay localized in the continuum, even in the absence of any defect [97], disorder [48], or nonlinearity [46]. Demonstrations of CLSs and a variety of associated phenomena have also been discussed in 1D and 2D lattices, including the Stub [39], sawtooth [34,91], rhombic [31,38,84,[98][99][100][101][102][103], Kagome [36,86], super-honeycomb [85,104], and Lieb [35,81] lattices, thanks to the advanced techniques that allow for precise flat-band engineering. Figure 4 shows some typical experimental results of CLSs observed in the Lieb, rhombic, and sawtooth lattices fabricated by the fs-laser-writing technique.…”

“…CLSs in Lieb lattices CLSs in rhombic lattices Localization in sawtooth lattices Shown are experimental results of (A) linear image propagation through a Lieb photonic lattice [35] and (B) a bound-state transmission in a pyrochlore-like (Kagome) photonic lattice [36]. (C) Experimental observation of a quincunx-shaped CLS which spans over two unit cells in a photonic rhombic lattice [84]. From left to right: the lattice, calculated band structure in the tight-binding approximation, probe beam input, in-phase output, and out-of-phase output.…”