Observation of linear and nonlinear light trapping on topological dislocations

Abstract

Topological dislocations in otherwise periodic lattices represent global structural defects that, nevertheless, typically leave the lattice periodicity intact far from the dislocation. Such dislocations arise in diverse physical systems ranging from crystalline solids, acoustic and photonic lattices and crystals to matter waves in optical lattices. Dislocations drastically affect the evolution of wave excitations in their vicinity, enabling novel mechanisms for trapping on topological defects and controlling the energy flow. Moreover, when combined with nonlinearity, such systems give rise to new types of self-sustained states of topological origin that have never been observed to date. Here we demonstrate experimentally, for the first time at optical frequencies, the waveguiding at various types of topological edge dislocations, resulting in the formation of localized photonic eigenstates with distinct and tunable shapes. Using femtosecond laser-writing techniques, we fabricated waveguide arrays with precisely tailored dislocation parameters, enabling full control over the degree of localization and internal structure of the associated modes. We further demonstrate both theoretically and experimentally that in the high-power regime, the families of thresholdless dislocation solitons bifurcate from such modes, which inherit shape diversity of their linear counterparts. Our results reveal a nontrivial interplay between nonlinearity and global lattice deformations and establish dislocation solitons as a new class of nonlinear topological states. They may stimulate the observation of new types of nonlinear states and interaction scenarios for excitations in nonlinear physical systems, where lattices with controllable global deformations can be created.

Figures (5)

• Figure 1: Eigenmodes of arrays with dislocation.(a–b) Microphotographs of the arrays with dislocations, where either two (a) or three (b) waveguide layers merge into one. The red horizontal lines are drawn in accordance with the value of the shift parameter $\delta_y$, which is larger in the left panels. Magenta dashed-dotted lines trace the waveguide positions within the arrays. The red arrows correspond to the Burgers vectors ${\bm \beta}$(c–d) Linear spectra of the dislocation arrays as a function of $\delta_y$ parameter for configurations with two (c) and three (d) merging layers. The color of the curves in (c-d) illustrates the form-factor of the eigenstates that quantifies their localization degree (red color corresponds to well-localized modes, while dark gray corresponds to extended states). Points marked with magenta stars in (c) correspond to eigenmodes shown in (e), while points marked with magenta diamonds in (d) correspond to eigenmodes in (f). Black circles in (e-f) indicate waveguide positions. Arrays and eigenmodes are displayed within the window $x,\,y\in[-250\,\mu \textrm{m},+250\,\mu \textrm{m}]$. Here and below $p=3.1$, $d=4.0$, $w_x=0.75$, and $w_y=0.65$.
• Figure 2: Families of solitons on dislocation. Power $U$ versus propagation constant $b$ for families of solitons (black lines) in arrays with dislocations with two (a) and three (c) merging layers. Panel (a) corresponds to $\delta_y=+3.2$, while panel (c) corresponds to $\delta_y=-1.6$. Gray region corresponds to the bulk band. Vertical dashed lines indicate the eigenvalue of the linear dislocation states from which thresholdless soliton families bifurcate. Red lines in (a) and (c) show perturbation growth rates for all depicted soliton families (the symbols in $\lambda_{\text{re}}(b)$ dependencies correspond to symbols on the respective soliton families). Profiles of solitons corresponding to violet dots in (a) and (c) are shown in (b) and (d), respectively. Black circles in (b) and (d) indicate positions of the waveguides in the array.
• Figure 3: Excitation of the in-phase dislocation solitons. Microphotographs of the waveguide arrays with dislocation (two merging layers) at $\delta_y=3.2$(a) and $\delta_y=0.8$(b). Red circles indicate the waveguides that were excited with in-phase beams. Output intensity distributions for different input peak powers $P$ in arrays with $\delta_y=3.2$(c) and $\delta_y=0.8$(d).
• Figure 4: Excitation of the out-of-phase dislocation solitons. Microphotographs of the waveguide arrays with dislocation (two merging layers) at $\delta_y=3.2$(a) and $\delta_y=0.8$(b). Red and blue circles indicate the waveguides that were excited with out-of-phase beams. Output intensity distributions for different input peak powers $P$ in arrays with $\delta_y=3.2$(c) and $\delta_y=0.8$(d).
• Figure 5: Excitation of the in-phase dislocation solitons on three merging layers. Microphotographs of the waveguide arrays with dislocation at $\delta_y=-1.6$(a) and $\delta_y=-4.0$(b), with red circles indicating the elliptical excitation of several central waveguides. Output intensity distributions for different input peak powers $P$ of elliptical beam in arrays $\delta_y=-1.6$(c) and $\delta_y=-4.0$(d).