Fig. 1. Scheme for the mode manipulation in SD assisted with ANNs. A probe beam at input representing one mode (edge in SD) is launched into different synthetic mode arrays, through which light is either transported laterally (left) or confined in SD but as a complex profile in real space (right), depending on the ANN design of the arrays. The input and output data of the ANNs are based on the preassigned eigenvalues and couplings of the arrays. Waveguides are curved along the z direction in real space. Vertical planes show the mode evolution in SD, where orange/purple profiles lined up vertically are the eigenmode distributions, forming the lattices in SD. The yellow bars denote the mode distribution of the probe beam, and the shaded zone (in right panel) represents a coupling blockade in SD. The curved lines depicted at the input and output facets of the arrays represent the complex beam profile in real space, which is well maintained during propagation in the right panel due to the proper design of the coupling blockade.

Fig. 2. Illustration of the mode evolution in different mode arrays designed by ANNs. (a1)–(a4) Illustration of the mode arrays with equal spacing of eigenvalues βm. (a1) The sketch of the eigenvalue array B and corresponding eigenmodes |φi⟩. The arrangement of the coupling array in real space (labeled T) is calculated by ANNs. (a2) The mode evolution dynamics in SD. Orange circle in the left column indicates the excited mode. (a3) The corresponding beam propagation dynamics in real space. (b1)–(b3) have the same layout as (a1)–(a3), except that they are for the mode arrays with outlying edges, showing that the excited mode is well confined in SD. The shaded zones in (b2) show the coupling blockades between the edge and bulk modes in SD. The propagation distances at the vertical lines in (a) and (b) are for Z=40mm and Z=80mm, respectively.

Fig. 3. Experimental demonstration of mode manipulation in SD and corresponding simulations. (a) Illustration of the cw-laser writing and cascade probing method in the experiment. Curved waveguide arrays are written section by section (from the top of a nonlinear crystal), and the output of the probe beam (propagating through the arrays along the z direction) from one section is taken as the input for the subsequent section assisted with the SLM, thus effectively increasing the propagation distance. (b) Results from the mode array with equal spacing, where (b1) and (b2) show the output amplitude and phase distribution from the experiment and simulation at z=20 and 40 mm. (b3) The corresponding output distribution in SD. (c1)–(c3) Results from the mode array with outlying edges, with the same layout as (b1)–(b3) but at even longer propagation distances, showing confinement of the excited mode in both real space and SD.

Fig. 4. Mode switching and morphing into topological modes by tuning the array in SD. (a) Mode switching between bulk modes in a topologically trivial lattice designed by ANNs. (a1) The lattice illustration in real space (far left column) and corresponding eigenvalue distribution (right panel). Bulk modes above or below the gap couple to each other without a coupling blockade under an array wiggling frequency Ω1=1 and the eigenvalue difference Δβ1=1. (a2) Mode evolution during propagation in SD, where the orange circle indicates the initially excited mode. The second region distinguished by the vertical lines is straight waveguides. The shaded zones indicate the coupling blockades in SDs in different regions. (a3) The light evolves in real space, where L1=36.3 is the propagation length in the first region. The plot on the right shows the average intensity distribution in the straight waveguide region. (b) Same layout as (a), but in a topologically nontrivial lattice showing the morphing of bulk modes into a zero-energy topological mode under an eigenvalue difference Δβ1′=Δβ1, Δβ2′=3Δβ1 and the wiggling frequency Ω1′=Ω1, Ω2′=3Ω1. The propagation length L1′=38.3 and L2′=4.22 in the first and second regions in (b3), respectively.