Topological Learning for the Classification of Disorder: An Application to the Design of Metasurfaces

Abstract

Structural disorder can improve the optical properties of metasurfaces, whether it is emerging from some large-scale fabrication methods or explicitly designed and built lithographically. For example, correlated disorder, induced by a minimum inter-nanostructure distance or by hyperuniformity properties, is particularly beneficial for light extraction. Inspired by topology, we introduce numerical descriptors to provide quantitative measures of disorder with universal properties, suitable to treat both uncorrelated and correlated disorder at all length scales. The accuracy of these topological descriptors is illustrated both theoretically and experimentally by using them to design plasmonic metasurfaces with controlled disorder that we then correlate to the strength of their surface lattice resonances. These descriptors are an example of topological tools that can be used for the fast and accurate design of disordered structures or as aid in improving their fabrication methods.

Keywords: design; disorder; metasurface; optimization; plasmonic; surface lattice resonance; topological data analysis.

Figure 1

Examples of randomly generated disordered lattices. The top and bottom row lattices are generated using S_d = 0.2 and S_d = 0.4, respectively. The lattices in the left column are uncorrelated, L_c = 0, while those in the middle and right columns have nonzero correlation, L_c = 6.

Figure 2

Examples of two key TDA processes used in this paper. The top row represents the computation of persistent homology from the data set (a) to its representation in a persistent diagram (d). In (b) and (c) are represented the circles whose diameters correspond respectively to the birth and death of the loop of this data set (single H₁ point in the persistence diagram in panel d). The bottom row represents the computation of the embedding of data sets (e) in a two-dimensional space (h) via the computation of their persistence diagrams (f) and the distance between them (g). Data sets of the same type are clustered in the embedding space (panel h).

Figure 3

Scatter plots of the two-dimensional embedding of three sets of generated lattices with uncorrelated (a), weakly correlated (b), and strongly correlated (c) disorder. Each set was generated from an original square lattice of period 500, 600, and 700 nm (left to right in panel a) and with S_d ∈ [0, 0.4]. In the absence of correlation, lattices with different values of the period and of S_d are well clustered. In the inset of panel a we adapted the size of points to illustrate how clustered the lattices are. The clustering is lost in the presence of correlations, panels b and c. Panels d and f and panels e and g are equivalent to panels a and c, respectively, with color coding based on the value of nSH₀ (TD) in panels d and f (e and g). In both cases the color gradient is not significantly affected by correlation.

Figure 4

Theoretical investigation of the correlation between TD and the strength of SLR. Panels a, b, and c represent respectively the generated metasurfaces of lowest, median, and highest TD. Their computed reflectance spectrum, in arbitrary units, is represented in d.

Figure 5

SEM images of the experimental samples (top row) and their transmittance spectra under normal incidence light linearly polarized parallel (middle row) or perpendicular (bottom row) to the long axis of the nanodisks. Each plot displays the spectra of a low and high TD metasurface, dashed light green and solid green, respectively, and an ordered metasurface with the same pitch (black). Each column corresponds to the metasurfaces generated with L_c ∈ [6, 8, 10] from left to right.

Figure 6

(a) Graph of the quality factors of the SLRs reported in Figure 5 in terms of the TD of the metasurfaces under normal incidence light linearly polarized parallel (green dots) and perpendicular (red crosses) to the long axis of the nanodisks. Two best-fit lines are added to represent the general trend of the quality factors for the parallel polarization, in green, and perpendicular polarization, in red, in terms of TD. (b) SEM image of the lattice with a high TD and a high quality factor, indicated by the arrow.