Identifying topology of leaky photonic lattices with machine learning

TL;DR

This work tackles identifying the topology of leaky photonic lattices from limited bulk intensity data, avoiding phase retrieval. It introduces a TBM-based dataset for a 1D dimerized SSH lattice with leaky channels and compares unsupervised (t-SNE) and supervised (CNN/MLP) learning, showing that supervised methods achieve high accuracy in classifying four edge configurations from fixed-$L$ intensity data. The paper demonstrates that network performance improves with propagation distance and central-window size, and that transfer learning can extend applicability to weak disorder, though accuracy degrades with stronger disorder. The results offer a practical route to infer bulk-boundary topology in nanophotonic systems under realistic measurement constraints, with potential extensions to Hamiltonian reconstruction under symmetry constraints.

Abstract

We show how machine learning techniques can be applied for the classification of topological phases in leaky photonic lattices using limited measurement data. We propose an approach based solely on bulk intensity measurements, thus exempt from the need for complicated phase retrieval procedures. In particular, we design a fully connected neural network that accurately determines topological properties from the output intensity distribution in dimerized waveguide arrays with leaky channels, after propagation of a spatially localized initial excitation at a finite distance, in a setting that closely emulates realistic experimental conditions.

Figures (8)

• Figure 1: (A) Schematic of a dimerized lattice of single-mode dielectric waveguides with tunable radiative losses and a possible experiment: the waveguide indexed by $i$ is excited at the input as indicated by a yellow circle, the intensity distribution is measured in the central area of $N_c$ elements at the output of the sample (the gray rectangle) to generate a dataset for learning the topological properties. (B) Tight binding model visualization of the photonic lattice in (A). The red and orange circles depict the main array -- a one-dimensional dimerised SSH-like array of coupled elements. Gray circles illustrate auxiliary arrays constituting leaky channels attached to the main array. The differing dashing between the elements denote different coupling strengths. (C) Band structures of the main (dashed red lines) and auxiliary (gray solid line) arrays in the designed leaky photonic lattice inscribed in glass. (D) Different configurations of the two edges in a finite lattice. (E) The output intensity distribution (colored) overlaid with the proposed lattice cross-section. (F,G) Intensity distribution, numerically obtained in paraxial modeling at the output facet of the waveguide array for (F) the Hermitian (lossless) lattice and (G) the lattice with leaky channels.
• Figure 2: (A,B) Evolution characteristics of the field in the main array in the lattice with fixed parameters obtained in the TBM of the nontrivial SSH array with (gold curves) and without (green curves) leaky channels. The Zak phase at $z>4$ cm converges to the quantised $\pi$ value, provided $N_{\text{env}}= 14$ elements in leaky channels. (C,D) Field evolution in $N$ elements of the main array assembled in a nontrivial (C) and trivial (D) configuration with fixed parameters of the lattice. The gray line on the right side marks the area of $N_c$ central waveguides, the intensity of which is fed to the input of the neural network.
• Figure 3: t-SNE maps of the system having 4 topological classes depending on its 2 edges: (A-C) Hermitian lattice, (D-F) lattice with leaky channels. The waveguide excited at the input is indexed by $i$. (A,B,D,E) correspond to the case of single-waveguide excitation: (A,D) $i=11$ is odd, (B,E) $i=12$ is even, (C,F) the excited waveguide is randomly chosen within a dimer. For each point in the two-dimensional parameter space there is a corresponding intensity distribution vector of dimension $N=22$ (or $N=23$), depending on the topological class. The four classes are color-coded: 00 (blue), 11 (red), 10 (green), 01 (black).
• Figure 4: Statistical characteristics of intensity distributions in waveguides. The datasets were prepared for the Hermitian (A) and leaky (B) cases assuming two possible positions $i=11,12$ of the initial excitation at $L=7.6$ cm. The mean value is indicated by markers in the middle of horizontal lines, while the standard deviation is represented by the borders of the lines. The classes are color-coded: 00 (blue squares), 11 (red circles), 01 (black right-facing triangles), 10 (green left-facing triangles). The total number of waveguides $N$ is 22 (even) for classes 00 and 11, and 23 (odd) for classes 01 an 10.
• Figure 5: (A) Accuracy of supervised learning methods as a function of the propagation distance $L$. (B) Scheme of the convolutional neural network, which takes the intensity distribution at $z=L$ as the input and determines topology of the lattice edges, $N_c=16$.